Preprints and abstracts: A. Boivin

Preprints and abstracts: André Boivin

List of Publications:

A full list of publications is available here.

Abstract of recents papers and preprints:

Abstract: Let F be a closed subset of C, let A_n(F) be the class of functions continuous on F and n-analytic (i.e. polynalytic of order n) on F⁰, and let E_n(F) (respectively M_n(F) denote the set of functions in C(F) that can be approximated uniformly on F by n-analytic entire functions (respectively by n-analytic meromorphic functions on C having no poles on F). We give several necessary or sufficient conditions on F, apparently independent of n, for A_n(F)=E_n(F) (respectively for A_n(F)=M_n(F) ) to hold. In particular, for compact sets F , our results considerably generalize the main results of J.J. Carmona, P.V. Paramonov and K.Yu. Fedorovski in Sbornik Math. 193 (2002), 1469-1492.

to be submitted (14 pages)

The dvi file for this article is available here

------ ------ ------

Completeness of systems of exponentials and Lambert W functions
by A. Boivin and H. Zhong

Abstract: We study some of the properties of the solution system {e^{i µ_nt}} of the delay-differential equation y'(t) = ay(t-1). We first establish some general results on the stability of the completeness of exponential systems in L² and then show that the solution system above is always complete, but is not a Schauder basis in L²(-½,½) if a = -1/e.

submitted (35 pages)

The dvi file for this article is available here

------ ------ ------

Bases of exponentials in weighted L^p spaces
by A. Boivin and A.M. Sedletskii

Abstract: We give some conditions on the generating function of the system of exponentials (e^{i µ_nt}) that guarantee that the system forms a basis in the spaces L^p((-a,a), (a-|t|)^bdt), p in (1, infty), b in [0, p-1). Previously, these theorems were knowns only for the case b = 0.

submitted (23 pages)

The dvi file for this article is available here

------ ------ ------

Uniform approximation by meromorphic functions on Riemann surfaces
by A. Boivin and B. Jiang

Abstract: A closed subset E of a Riemann surface S is called a set of uniform meromorphic approximation if every function f continuous on E and holomorphic on E⁰ can be approximated uniformly on E by meromorphic functions on S. We show that if E is a set of meromorphic approximation, then so is E \cap D for every compact parametric disk D. As a consequence, we obtain a generalization to Riemann surfaces of a well-known theorem of A.G. Vitushkin characterizing the compact sets of uniform meromorphic approximation. More precisely we prove that if K is a compact subset of an arbitrary Riemann surface then A(K) = H(K) if and only if for every point p in K, there exists a closed parametric disk D centred at p such that A(K \cap D) = H(K \cap D).

Journal d'Analyse Mathématique (to appear -- 19 pages)

The dvi file for this article is available here

------ ------ ------

The growth of an entire function and its Dirichlet coefficients and exponents.
by A. Boivin and C. Zhu

Abstract: Let F be an entire function represented by a (generalized) Dirichlet series F(s) = sum d_ne^{-t_n s} where the coefficients {d_n} and the exponents {t_n} are sequences of complex numbers. We introduce a modified (R)-order rho and modified (R)-type sigma and we obtain an estimate for |d_n| when n is sufficiently large in terms of rho, sigma and t_n. Other estimates relating rho and sigma to d_n and t_n are also obtained.

Complex Variables - Theory and Application, 48, 397-416 (2003)

The dvi file for this article is available here

------ ------ ------

On the completeness of the system { z ^t_n} in L².
by A. Boivin and C. Zhu

Abstract: Given an unbounded domain G located outside an angle domain with vertex at the origin, and a sequence of distinct complex numbers {t_n} satisfying n/|t_n| tends to D as n tends to infinity with D positive and finite, and |arg(t_n)| < A < pi/2, we obtain a completeness theorem for the system { z^t_n} in L²_a. The case with weight is also considered.
© 2002 Academic Press Inc.

Journal of Approximation Theory 118, 1-19 (2002)

------ ------ ------

Approximation on closed sets by analytic or meromorphic solutions
of elliptic equations and applications
by A. Boivin, P.M. Gauthier and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant complex coefficients and a class of functions (jet-distributions) which are defined on a (relatively) closed subset of a domain G in Rⁿ and which belong locally to a Banach space V, we consider the problem of approximating in the norm of V the functions in this class by ``analytic" and ``meromorphic" solutions of the equation Lu=0. We establish new Roth, Arakelyan (including tangential) and Carleman type theorems for a large class of Banach spaces V and operators L. Important applications to boundary value problems of solutions of homogeneous elliptic partial differential equations are obtained, including the solution of a generalized Dirichlet problem.

Canadian Mathematics Journal 54, 945-969 (2002)

------ ------ ------

On the L²-completeness of some systems of analytic functions
by A. Boivin and C. Zhu

Abstract: For an entire function f(z), a sequence of complex numbers {µ_n} and an unbounded domain G, the completeness of the system {f(µ_nz)} in L²_a(G) (mean-square approximation) is studied. A similar problem is discussed when f is analytic on the Riemann surface of the logarithm.
© 2001 OPA (Overseas Publishers Association) N.V.

Complex Variables - Theory and Application, 45, 273-295 (2001)

------ ------ ------

Holomorphic and harmonic approximation on Riemann surfaces
by A. Boivin and P.M. Gauthier

Abstract: The most important result on open Riemann surfaces is the analog of Runge's theorem on approximation by complex polynomials, due to Behnke and Stein. In this course, we begin by a discussion of Runge's theorems on polynomial and rational approximation on compact sets. This theory is refined and extended in various ways to Riemann surfaces. We also introduce a corresponding theory of harmonic approximation.
© 2001 Kluwer Academic Publishers

Approximation, Complex Analysis and Potential Theory
Proceedings of a NATO ASI held in Montreal, July 2000
N. Arakelian and P.M. Gauthier, eds.
NATO Science Series II, Vol. 37
Kluwer Academic Publishers 2001, 107--128

------ ------ ------

On radial limit functions for entire solutions of second order elliptic equations in R²
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L of order two with constant complex coefficients in R², we consider entire solutions of the equation Lu = 0 for which U(e^it) := lim u(re^it) exists for all t in [0, 2pi) as a finite limit in C as r goes to infinity. We characterize the possible ``radial limit functions" U. This is an analog of the work of Alice Roth for entire holomorphic functions. The results seem new even for harmonic functions.

Publicacions Matemàtiques, 42, No.2, 509-519 (1998)

The dvi file for this article is available here

------ ------ ------

Approximation by meromorphic and entire solutions
of elliptic equations in Banach spaces of distributions
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant coefficients and a class of functions (jets-distributions) which are defined on a closed, not necessarily compact, subset of Rⁿ and which belong locally to a Banach space V, we consider the approximation in the norm of V of the functions (jets) in this class by global (entire or meromorphic) solutions of the equation Lu=0. We establish Runge, Mergelyan, Roth and Arakelyan - type theorems for a large class of Banach spaces V and operators L, encompassing most of the previously considered generalizations of these theorems and obtaining new ones.

Sbornik Math. 189, No.4, 481--502 (1998) (in English)
Mat. Sbornik 189, No.4, 3--24 (1998) (in Russian)

The dvi file for this article (in English) is available here

------ ------ ------

Approximation by entire functions
by A. Boivin and P.V. Paramonov

Abstract: Examples of the type of (new) results we can deduce from the theorems obtained in the paper "Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions" are given in the case when L is the Cauchy-Riemann or the Laplacian operator and the norm is the C^m norm.

Proceeding of a Conference in memory of A.F. Leont'ev, Nizhni Novgorod, Russia, June 2-5, 1997, 91-92.

The dvi file for this article is available here

------ ------ ------

Extension and Approximation of CR functions on tube manifolds
by A. Boivin and R. Dwilewicz

Abstract: In this paper we prove the following theorem: Any continuous CR function on a tube manifold t(N) = N x iRⁿ in Cⁿ, where N is an embedded submanifold of Rⁿ of class C², can be continuously CR extended to the tube over the convex hull of N. There are no additional assumptions on N, and in this sense, the theorem generalizes all known results of this type, including the classical Bochner tube theorem. Also it implies almost uniform approximation of CR functions on tubes by holomorphic polynomials in Cⁿ.

Trans. Amer. Math. Soc. 350, 1945-1956 (1998)

The dvi file for this article is available here

------ ------ ------

An example in tangential meromorphic approximation
by A. Boivin and A.H. Nersessian

Abstract: It is known that all sets of meromorphic uniform approximation in C which satisfy an additional condition involving the Gleason parts of the algebra R(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, though it is a set of tangential approximation, does not satisfy this extra condition on parts, and thus showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem.
© 1996 Academic Press Inc.

Journal of Approximation Theory 87, 103-111 (1996)

------ ------ ------

Meromorphic approximation in weighted L^p spaces
by A. Boivin, A. Bonilla and J.C. Fariña

Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995)
------ ------ ------

Completeness of systems of exponentials and Lambert W functions
by A. Boivin and H. Zhong

Abstract: We study some of the properties of the solution system {e^{i µ_nt}} of the delay-differential equation y'(t) = ay(t-1). We first establish some general results on the stability of the completeness of exponential systems in L² and then show that the solution system above is always complete, but is not a Schauder basis in L²(-½,½) if a = -1/e.

submitted (35 pages)

The dvi file for this article is available here

------ ------ ------

Bases of exponentials in weighted L^p spaces
by A. Boivin and A.M. Sedletskii

Abstract: We give some conditions on the generating function of the system of exponentials (e^{i µ_nt}) that guarantee that the system forms a basis in the spaces L^p((-a,a), (a-|t|)^bdt), p in (1, infty), b in [0, p-1). Previously, these theorems were knowns only for the case b = 0.

submitted (23 pages)

The dvi file for this article is available here

------ ------ ------

Uniform approximation by meromorphic functions on Riemann surfaces
by A. Boivin and B. Jiang

Abstract: A closed subset E of a Riemann surface S is called a set of uniform meromorphic approximation if every function f continuous on E and holomorphic on E⁰ can be approximated uniformly on E by meromorphic functions on S. We show that if E is a set of meromorphic approximation, then so is E \cap D for every compact parametric disk D. As a consequence, we obtain a generalization to Riemann surfaces of a well-known theorem of A.G. Vitushkin characterizing the compact sets of uniform meromorphic approximation. More precisely we prove that if K is a compact subset of an arbitrary Riemann surface then A(K) = H(K) if and only if for every point p in K, there exists a closed parametric disk D centred at p such that A(K \cap D) = H(K \cap D).

Journal d'Analyse Mathématique (to appear -- 19 pages)

The dvi file for this article is available here

------ ------ ------

The growth of an entire function and its Dirichlet coefficients and exponents.
by A. Boivin and C. Zhu

Abstract: Let F be an entire function represented by a (generalized) Dirichlet series F(s) = sum d_ne^{-t_n s} where the coefficients {d_n} and the exponents {t_n} are sequences of complex numbers. We introduce a modified (R)-order rho and modified (R)-type sigma and we obtain an estimate for |d_n| when n is sufficiently large in terms of rho, sigma and t_n. Other estimates relating rho and sigma to d_n and t_n are also obtained.

Complex Variables - Theory and Application, 48, 397-416 (2003)

The dvi file for this article is available here

------ ------ ------

On the completeness of the system { z ^t_n} in L².
by A. Boivin and C. Zhu

Abstract: Given an unbounded domain G located outside an angle domain with vertex at the origin, and a sequence of distinct complex numbers {t_n} satisfying n/|t_n| tends to D as n tends to infinity with D positive and finite, and |arg(t_n)| < A < pi/2, we obtain a completeness theorem for the system { z^t_n} in L²_a. The case with weight is also considered.
© 2002 Academic Press Inc.

Journal of Approximation Theory 118, 1-19 (2002)

------ ------ ------

Approximation on closed sets by analytic or meromorphic solutions
of elliptic equations and applications
by A. Boivin, P.M. Gauthier and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant complex coefficients and a class of functions (jet-distributions) which are defined on a (relatively) closed subset of a domain G in Rⁿ and which belong locally to a Banach space V, we consider the problem of approximating in the norm of V the functions in this class by ``analytic" and ``meromorphic" solutions of the equation Lu=0. We establish new Roth, Arakelyan (including tangential) and Carleman type theorems for a large class of Banach spaces V and operators L. Important applications to boundary value problems of solutions of homogeneous elliptic partial differential equations are obtained, including the solution of a generalized Dirichlet problem.

Canadian Mathematics Journal 54, 945-969 (2002)

------ ------ ------

On the L²-completeness of some systems of analytic functions
by A. Boivin and C. Zhu

Abstract: For an entire function f(z), a sequence of complex numbers {µ_n} and an unbounded domain G, the completeness of the system {f(µ_nz)} in L²_a(G) (mean-square approximation) is studied. A similar problem is discussed when f is analytic on the Riemann surface of the logarithm.
© 2001 OPA (Overseas Publishers Association) N.V.

Complex Variables - Theory and Application, 45, 273-295 (2001)

------ ------ ------

Holomorphic and harmonic approximation on Riemann surfaces
by A. Boivin and P.M. Gauthier

Abstract: The most important result on open Riemann surfaces is the analog of Runge's theorem on approximation by complex polynomials, due to Behnke and Stein. In this course, we begin by a discussion of Runge's theorems on polynomial and rational approximation on compact sets. This theory is refined and extended in various ways to Riemann surfaces. We also introduce a corresponding theory of harmonic approximation.
© 2001 Kluwer Academic Publishers

Approximation, Complex Analysis and Potential Theory
Proceedings of a NATO ASI held in Montreal, July 2000
N. Arakelian and P.M. Gauthier, eds.
NATO Science Series II, Vol. 37
Kluwer Academic Publishers 2001, 107--128

------ ------ ------

On radial limit functions for entire solutions of second order elliptic equations in R²
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L of order two with constant complex coefficients in R², we consider entire solutions of the equation Lu = 0 for which U(e^it) := lim u(re^it) exists for all t in [0, 2pi) as a finite limit in C as r goes to infinity. We characterize the possible ``radial limit functions" U. This is an analog of the work of Alice Roth for entire holomorphic functions. The results seem new even for harmonic functions.

Publicacions Matemàtiques, 42, No.2, 509-519 (1998)

The dvi file for this article is available here

------ ------ ------

Approximation by meromorphic and entire solutions
of elliptic equations in Banach spaces of distributions
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant coefficients and a class of functions (jets-distributions) which are defined on a closed, not necessarily compact, subset of Rⁿ and which belong locally to a Banach space V, we consider the approximation in the norm of V of the functions (jets) in this class by global (entire or meromorphic) solutions of the equation Lu=0. We establish Runge, Mergelyan, Roth and Arakelyan - type theorems for a large class of Banach spaces V and operators L, encompassing most of the previously considered generalizations of these theorems and obtaining new ones.

Sbornik Math. 189, No.4, 481--502 (1998) (in English)
Mat. Sbornik 189, No.4, 3--24 (1998) (in Russian)

The dvi file for this article (in English) is available here

------ ------ ------

Approximation by entire functions
by A. Boivin and P.V. Paramonov

Abstract: Examples of the type of (new) results we can deduce from the theorems obtained in the paper "Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions" are given in the case when L is the Cauchy-Riemann or the Laplacian operator and the norm is the C^m norm.

Proceeding of a Conference in memory of A.F. Leont'ev, Nizhni Novgorod, Russia, June 2-5, 1997, 91-92.

The dvi file for this article is available here

------ ------ ------

Extension and Approximation of CR functions on tube manifolds
by A. Boivin and R. Dwilewicz

Abstract: In this paper we prove the following theorem: Any continuous CR function on a tube manifold t(N) = N x iRⁿ in Cⁿ, where N is an embedded submanifold of Rⁿ of class C², can be continuously CR extended to the tube over the convex hull of N. There are no additional assumptions on N, and in this sense, the theorem generalizes all known results of this type, including the classical Bochner tube theorem. Also it implies almost uniform approximation of CR functions on tubes by holomorphic polynomials in Cⁿ.

Trans. Amer. Math. Soc. 350, 1945-1956 (1998)

The dvi file for this article is available here

------ ------ ------

An example in tangential meromorphic approximation
by A. Boivin and A.H. Nersessian

Abstract: It is known that all sets of meromorphic uniform approximation in C which satisfy an additional condition involving the Gleason parts of the algebra R(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, though it is a set of tangential approximation, does not satisfy this extra condition on parts, and thus showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem.
© 1996 Academic Press Inc.

Journal of Approximation Theory 87, 103-111 (1996)

------ ------ ------

Meromorphic approximation in weighted L^p spaces
by A. Boivin, A. Bonilla and J.C. Fariña

Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995): Abstract: We study some of the properties of the solution system {e^{i µ_nt}} of the delay-differential equation y'(t) = ay(t-1). We first establish some general results on the stability of the completeness of exponential systems in L² and then show that the solution system above is always complete, but is not a Schauder basis in L²(-½,½) if a = -1/e.

submitted (35 pages)

The dvi file for this article is available here

------ ------ ------

Bases of exponentials in weighted L^p spaces
by A. Boivin and A.M. Sedletskii

Abstract: We give some conditions on the generating function of the system of exponentials (e^{i µ_nt}) that guarantee that the system forms a basis in the spaces L^p((-a,a), (a-|t|)^bdt), p in (1, infty), b in [0, p-1). Previously, these theorems were knowns only for the case b = 0.

submitted (23 pages)

The dvi file for this article is available here

------ ------ ------

Uniform approximation by meromorphic functions on Riemann surfaces
by A. Boivin and B. Jiang

Abstract: A closed subset E of a Riemann surface S is called a set of uniform meromorphic approximation if every function f continuous on E and holomorphic on E⁰ can be approximated uniformly on E by meromorphic functions on S. We show that if E is a set of meromorphic approximation, then so is E \cap D for every compact parametric disk D. As a consequence, we obtain a generalization to Riemann surfaces of a well-known theorem of A.G. Vitushkin characterizing the compact sets of uniform meromorphic approximation. More precisely we prove that if K is a compact subset of an arbitrary Riemann surface then A(K) = H(K) if and only if for every point p in K, there exists a closed parametric disk D centred at p such that A(K \cap D) = H(K \cap D).

Journal d'Analyse Mathématique (to appear -- 19 pages)

The dvi file for this article is available here

------ ------ ------

The growth of an entire function and its Dirichlet coefficients and exponents.
by A. Boivin and C. Zhu

Abstract: Let F be an entire function represented by a (generalized) Dirichlet series F(s) = sum d_ne^{-t_n s} where the coefficients {d_n} and the exponents {t_n} are sequences of complex numbers. We introduce a modified (R)-order rho and modified (R)-type sigma and we obtain an estimate for |d_n| when n is sufficiently large in terms of rho, sigma and t_n. Other estimates relating rho and sigma to d_n and t_n are also obtained.

Complex Variables - Theory and Application, 48, 397-416 (2003)

The dvi file for this article is available here

------ ------ ------

On the completeness of the system { z ^t_n} in L².
by A. Boivin and C. Zhu

Abstract: Given an unbounded domain G located outside an angle domain with vertex at the origin, and a sequence of distinct complex numbers {t_n} satisfying n/|t_n| tends to D as n tends to infinity with D positive and finite, and |arg(t_n)| < A < pi/2, we obtain a completeness theorem for the system { z^t_n} in L²_a. The case with weight is also considered.
© 2002 Academic Press Inc.

Journal of Approximation Theory 118, 1-19 (2002)

------ ------ ------

Approximation on closed sets by analytic or meromorphic solutions
of elliptic equations and applications
by A. Boivin, P.M. Gauthier and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant complex coefficients and a class of functions (jet-distributions) which are defined on a (relatively) closed subset of a domain G in Rⁿ and which belong locally to a Banach space V, we consider the problem of approximating in the norm of V the functions in this class by ``analytic" and ``meromorphic" solutions of the equation Lu=0. We establish new Roth, Arakelyan (including tangential) and Carleman type theorems for a large class of Banach spaces V and operators L. Important applications to boundary value problems of solutions of homogeneous elliptic partial differential equations are obtained, including the solution of a generalized Dirichlet problem.

Canadian Mathematics Journal 54, 945-969 (2002)

------ ------ ------

On the L²-completeness of some systems of analytic functions
by A. Boivin and C. Zhu

Abstract: For an entire function f(z), a sequence of complex numbers {µ_n} and an unbounded domain G, the completeness of the system {f(µ_nz)} in L²_a(G) (mean-square approximation) is studied. A similar problem is discussed when f is analytic on the Riemann surface of the logarithm.
© 2001 OPA (Overseas Publishers Association) N.V.

Complex Variables - Theory and Application, 45, 273-295 (2001)

------ ------ ------

Holomorphic and harmonic approximation on Riemann surfaces
by A. Boivin and P.M. Gauthier

Abstract: The most important result on open Riemann surfaces is the analog of Runge's theorem on approximation by complex polynomials, due to Behnke and Stein. In this course, we begin by a discussion of Runge's theorems on polynomial and rational approximation on compact sets. This theory is refined and extended in various ways to Riemann surfaces. We also introduce a corresponding theory of harmonic approximation.
© 2001 Kluwer Academic Publishers

Approximation, Complex Analysis and Potential Theory
Proceedings of a NATO ASI held in Montreal, July 2000
N. Arakelian and P.M. Gauthier, eds.
NATO Science Series II, Vol. 37
Kluwer Academic Publishers 2001, 107--128

------ ------ ------

On radial limit functions for entire solutions of second order elliptic equations in R²
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L of order two with constant complex coefficients in R², we consider entire solutions of the equation Lu = 0 for which U(e^it) := lim u(re^it) exists for all t in [0, 2pi) as a finite limit in C as r goes to infinity. We characterize the possible ``radial limit functions" U. This is an analog of the work of Alice Roth for entire holomorphic functions. The results seem new even for harmonic functions.

Publicacions Matemàtiques, 42, No.2, 509-519 (1998)

The dvi file for this article is available here

------ ------ ------

Approximation by meromorphic and entire solutions
of elliptic equations in Banach spaces of distributions
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant coefficients and a class of functions (jets-distributions) which are defined on a closed, not necessarily compact, subset of Rⁿ and which belong locally to a Banach space V, we consider the approximation in the norm of V of the functions (jets) in this class by global (entire or meromorphic) solutions of the equation Lu=0. We establish Runge, Mergelyan, Roth and Arakelyan - type theorems for a large class of Banach spaces V and operators L, encompassing most of the previously considered generalizations of these theorems and obtaining new ones.

Sbornik Math. 189, No.4, 481--502 (1998) (in English)
Mat. Sbornik 189, No.4, 3--24 (1998) (in Russian)

The dvi file for this article (in English) is available here

------ ------ ------

Approximation by entire functions
by A. Boivin and P.V. Paramonov

Abstract: Examples of the type of (new) results we can deduce from the theorems obtained in the paper "Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions" are given in the case when L is the Cauchy-Riemann or the Laplacian operator and the norm is the C^m norm.

Proceeding of a Conference in memory of A.F. Leont'ev, Nizhni Novgorod, Russia, June 2-5, 1997, 91-92.

The dvi file for this article is available here

------ ------ ------

Extension and Approximation of CR functions on tube manifolds
by A. Boivin and R. Dwilewicz

Abstract: In this paper we prove the following theorem: Any continuous CR function on a tube manifold t(N) = N x iRⁿ in Cⁿ, where N is an embedded submanifold of Rⁿ of class C², can be continuously CR extended to the tube over the convex hull of N. There are no additional assumptions on N, and in this sense, the theorem generalizes all known results of this type, including the classical Bochner tube theorem. Also it implies almost uniform approximation of CR functions on tubes by holomorphic polynomials in Cⁿ.

Trans. Amer. Math. Soc. 350, 1945-1956 (1998)

The dvi file for this article is available here

------ ------ ------

An example in tangential meromorphic approximation
by A. Boivin and A.H. Nersessian

Abstract: It is known that all sets of meromorphic uniform approximation in C which satisfy an additional condition involving the Gleason parts of the algebra R(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, though it is a set of tangential approximation, does not satisfy this extra condition on parts, and thus showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem.
© 1996 Academic Press Inc.

Journal of Approximation Theory 87, 103-111 (1996)

------ ------ ------

Meromorphic approximation in weighted L^p spaces
by A. Boivin, A. Bonilla and J.C. Fariña

Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995)
------ ------ ------

Bases of exponentials in weighted L^p spaces
by A. Boivin and A.M. Sedletskii

Abstract: We give some conditions on the generating function of the system of exponentials (e^{i µ_nt}) that guarantee that the system forms a basis in the spaces L^p((-a,a), (a-|t|)^bdt), p in (1, infty), b in [0, p-1). Previously, these theorems were knowns only for the case b = 0.

submitted (23 pages)

The dvi file for this article is available here

------ ------ ------

Uniform approximation by meromorphic functions on Riemann surfaces
by A. Boivin and B. Jiang

Abstract: A closed subset E of a Riemann surface S is called a set of uniform meromorphic approximation if every function f continuous on E and holomorphic on E⁰ can be approximated uniformly on E by meromorphic functions on S. We show that if E is a set of meromorphic approximation, then so is E \cap D for every compact parametric disk D. As a consequence, we obtain a generalization to Riemann surfaces of a well-known theorem of A.G. Vitushkin characterizing the compact sets of uniform meromorphic approximation. More precisely we prove that if K is a compact subset of an arbitrary Riemann surface then A(K) = H(K) if and only if for every point p in K, there exists a closed parametric disk D centred at p such that A(K \cap D) = H(K \cap D).

Journal d'Analyse Mathématique (to appear -- 19 pages)

The dvi file for this article is available here

------ ------ ------

The growth of an entire function and its Dirichlet coefficients and exponents.
by A. Boivin and C. Zhu

Abstract: Let F be an entire function represented by a (generalized) Dirichlet series F(s) = sum d_ne^{-t_n s} where the coefficients {d_n} and the exponents {t_n} are sequences of complex numbers. We introduce a modified (R)-order rho and modified (R)-type sigma and we obtain an estimate for |d_n| when n is sufficiently large in terms of rho, sigma and t_n. Other estimates relating rho and sigma to d_n and t_n are also obtained.

Complex Variables - Theory and Application, 48, 397-416 (2003)

The dvi file for this article is available here

------ ------ ------

On the completeness of the system { z ^t_n} in L².
by A. Boivin and C. Zhu

Abstract: Given an unbounded domain G located outside an angle domain with vertex at the origin, and a sequence of distinct complex numbers {t_n} satisfying n/|t_n| tends to D as n tends to infinity with D positive and finite, and |arg(t_n)| < A < pi/2, we obtain a completeness theorem for the system { z^t_n} in L²_a. The case with weight is also considered.
© 2002 Academic Press Inc.

Journal of Approximation Theory 118, 1-19 (2002)

------ ------ ------

Approximation on closed sets by analytic or meromorphic solutions
of elliptic equations and applications
by A. Boivin, P.M. Gauthier and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant complex coefficients and a class of functions (jet-distributions) which are defined on a (relatively) closed subset of a domain G in Rⁿ and which belong locally to a Banach space V, we consider the problem of approximating in the norm of V the functions in this class by ``analytic" and ``meromorphic" solutions of the equation Lu=0. We establish new Roth, Arakelyan (including tangential) and Carleman type theorems for a large class of Banach spaces V and operators L. Important applications to boundary value problems of solutions of homogeneous elliptic partial differential equations are obtained, including the solution of a generalized Dirichlet problem.

Canadian Mathematics Journal 54, 945-969 (2002)

------ ------ ------

On the L²-completeness of some systems of analytic functions
by A. Boivin and C. Zhu

Abstract: For an entire function f(z), a sequence of complex numbers {µ_n} and an unbounded domain G, the completeness of the system {f(µ_nz)} in L²_a(G) (mean-square approximation) is studied. A similar problem is discussed when f is analytic on the Riemann surface of the logarithm.
© 2001 OPA (Overseas Publishers Association) N.V.

Complex Variables - Theory and Application, 45, 273-295 (2001)

------ ------ ------

Holomorphic and harmonic approximation on Riemann surfaces
by A. Boivin and P.M. Gauthier

Abstract: The most important result on open Riemann surfaces is the analog of Runge's theorem on approximation by complex polynomials, due to Behnke and Stein. In this course, we begin by a discussion of Runge's theorems on polynomial and rational approximation on compact sets. This theory is refined and extended in various ways to Riemann surfaces. We also introduce a corresponding theory of harmonic approximation.
© 2001 Kluwer Academic Publishers

Approximation, Complex Analysis and Potential Theory
Proceedings of a NATO ASI held in Montreal, July 2000
N. Arakelian and P.M. Gauthier, eds.
NATO Science Series II, Vol. 37
Kluwer Academic Publishers 2001, 107--128

------ ------ ------

On radial limit functions for entire solutions of second order elliptic equations in R²
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L of order two with constant complex coefficients in R², we consider entire solutions of the equation Lu = 0 for which U(e^it) := lim u(re^it) exists for all t in [0, 2pi) as a finite limit in C as r goes to infinity. We characterize the possible ``radial limit functions" U. This is an analog of the work of Alice Roth for entire holomorphic functions. The results seem new even for harmonic functions.

Publicacions Matemàtiques, 42, No.2, 509-519 (1998)

The dvi file for this article is available here

------ ------ ------

Approximation by meromorphic and entire solutions
of elliptic equations in Banach spaces of distributions
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant coefficients and a class of functions (jets-distributions) which are defined on a closed, not necessarily compact, subset of Rⁿ and which belong locally to a Banach space V, we consider the approximation in the norm of V of the functions (jets) in this class by global (entire or meromorphic) solutions of the equation Lu=0. We establish Runge, Mergelyan, Roth and Arakelyan - type theorems for a large class of Banach spaces V and operators L, encompassing most of the previously considered generalizations of these theorems and obtaining new ones.

Sbornik Math. 189, No.4, 481--502 (1998) (in English)
Mat. Sbornik 189, No.4, 3--24 (1998) (in Russian)

The dvi file for this article (in English) is available here

------ ------ ------

Approximation by entire functions
by A. Boivin and P.V. Paramonov

Abstract: Examples of the type of (new) results we can deduce from the theorems obtained in the paper "Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions" are given in the case when L is the Cauchy-Riemann or the Laplacian operator and the norm is the C^m norm.

Proceeding of a Conference in memory of A.F. Leont'ev, Nizhni Novgorod, Russia, June 2-5, 1997, 91-92.

The dvi file for this article is available here

------ ------ ------

Extension and Approximation of CR functions on tube manifolds
by A. Boivin and R. Dwilewicz

Abstract: In this paper we prove the following theorem: Any continuous CR function on a tube manifold t(N) = N x iRⁿ in Cⁿ, where N is an embedded submanifold of Rⁿ of class C², can be continuously CR extended to the tube over the convex hull of N. There are no additional assumptions on N, and in this sense, the theorem generalizes all known results of this type, including the classical Bochner tube theorem. Also it implies almost uniform approximation of CR functions on tubes by holomorphic polynomials in Cⁿ.

Trans. Amer. Math. Soc. 350, 1945-1956 (1998)

The dvi file for this article is available here

------ ------ ------

An example in tangential meromorphic approximation
by A. Boivin and A.H. Nersessian

Abstract: It is known that all sets of meromorphic uniform approximation in C which satisfy an additional condition involving the Gleason parts of the algebra R(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, though it is a set of tangential approximation, does not satisfy this extra condition on parts, and thus showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem.
© 1996 Academic Press Inc.

Journal of Approximation Theory 87, 103-111 (1996)

------ ------ ------

Meromorphic approximation in weighted L^p spaces
by A. Boivin, A. Bonilla and J.C. Fariña

Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995): Abstract: We give some conditions on the generating function of the system of exponentials (e^{i µ_nt}) that guarantee that the system forms a basis in the spaces L^p((-a,a), (a-|t|)^bdt), p in (1, infty), b in [0, p-1). Previously, these theorems were knowns only for the case b = 0.

submitted (23 pages)

The dvi file for this article is available here

------ ------ ------

Uniform approximation by meromorphic functions on Riemann surfaces
by A. Boivin and B. Jiang

Abstract: A closed subset E of a Riemann surface S is called a set of uniform meromorphic approximation if every function f continuous on E and holomorphic on E⁰ can be approximated uniformly on E by meromorphic functions on S. We show that if E is a set of meromorphic approximation, then so is E \cap D for every compact parametric disk D. As a consequence, we obtain a generalization to Riemann surfaces of a well-known theorem of A.G. Vitushkin characterizing the compact sets of uniform meromorphic approximation. More precisely we prove that if K is a compact subset of an arbitrary Riemann surface then A(K) = H(K) if and only if for every point p in K, there exists a closed parametric disk D centred at p such that A(K \cap D) = H(K \cap D).

Journal d'Analyse Mathématique (to appear -- 19 pages)

The dvi file for this article is available here

------ ------ ------

The growth of an entire function and its Dirichlet coefficients and exponents.
by A. Boivin and C. Zhu

Abstract: Let F be an entire function represented by a (generalized) Dirichlet series F(s) = sum d_ne^{-t_n s} where the coefficients {d_n} and the exponents {t_n} are sequences of complex numbers. We introduce a modified (R)-order rho and modified (R)-type sigma and we obtain an estimate for |d_n| when n is sufficiently large in terms of rho, sigma and t_n. Other estimates relating rho and sigma to d_n and t_n are also obtained.

Complex Variables - Theory and Application, 48, 397-416 (2003)

The dvi file for this article is available here

------ ------ ------

On the completeness of the system { z ^t_n} in L².
by A. Boivin and C. Zhu

Abstract: Given an unbounded domain G located outside an angle domain with vertex at the origin, and a sequence of distinct complex numbers {t_n} satisfying n/|t_n| tends to D as n tends to infinity with D positive and finite, and |arg(t_n)| < A < pi/2, we obtain a completeness theorem for the system { z^t_n} in L²_a. The case with weight is also considered.
© 2002 Academic Press Inc.

Journal of Approximation Theory 118, 1-19 (2002)

------ ------ ------

Approximation on closed sets by analytic or meromorphic solutions
of elliptic equations and applications
by A. Boivin, P.M. Gauthier and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant complex coefficients and a class of functions (jet-distributions) which are defined on a (relatively) closed subset of a domain G in Rⁿ and which belong locally to a Banach space V, we consider the problem of approximating in the norm of V the functions in this class by ``analytic" and ``meromorphic" solutions of the equation Lu=0. We establish new Roth, Arakelyan (including tangential) and Carleman type theorems for a large class of Banach spaces V and operators L. Important applications to boundary value problems of solutions of homogeneous elliptic partial differential equations are obtained, including the solution of a generalized Dirichlet problem.

Canadian Mathematics Journal 54, 945-969 (2002)

------ ------ ------

On the L²-completeness of some systems of analytic functions
by A. Boivin and C. Zhu

Abstract: For an entire function f(z), a sequence of complex numbers {µ_n} and an unbounded domain G, the completeness of the system {f(µ_nz)} in L²_a(G) (mean-square approximation) is studied. A similar problem is discussed when f is analytic on the Riemann surface of the logarithm.
© 2001 OPA (Overseas Publishers Association) N.V.

Complex Variables - Theory and Application, 45, 273-295 (2001)

------ ------ ------

Holomorphic and harmonic approximation on Riemann surfaces
by A. Boivin and P.M. Gauthier

Abstract: The most important result on open Riemann surfaces is the analog of Runge's theorem on approximation by complex polynomials, due to Behnke and Stein. In this course, we begin by a discussion of Runge's theorems on polynomial and rational approximation on compact sets. This theory is refined and extended in various ways to Riemann surfaces. We also introduce a corresponding theory of harmonic approximation.
© 2001 Kluwer Academic Publishers

Approximation, Complex Analysis and Potential Theory
Proceedings of a NATO ASI held in Montreal, July 2000
N. Arakelian and P.M. Gauthier, eds.
NATO Science Series II, Vol. 37
Kluwer Academic Publishers 2001, 107--128

------ ------ ------

On radial limit functions for entire solutions of second order elliptic equations in R²
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L of order two with constant complex coefficients in R², we consider entire solutions of the equation Lu = 0 for which U(e^it) := lim u(re^it) exists for all t in [0, 2pi) as a finite limit in C as r goes to infinity. We characterize the possible ``radial limit functions" U. This is an analog of the work of Alice Roth for entire holomorphic functions. The results seem new even for harmonic functions.

Publicacions Matemàtiques, 42, No.2, 509-519 (1998)

The dvi file for this article is available here

------ ------ ------

Approximation by meromorphic and entire solutions
of elliptic equations in Banach spaces of distributions
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant coefficients and a class of functions (jets-distributions) which are defined on a closed, not necessarily compact, subset of Rⁿ and which belong locally to a Banach space V, we consider the approximation in the norm of V of the functions (jets) in this class by global (entire or meromorphic) solutions of the equation Lu=0. We establish Runge, Mergelyan, Roth and Arakelyan - type theorems for a large class of Banach spaces V and operators L, encompassing most of the previously considered generalizations of these theorems and obtaining new ones.

Sbornik Math. 189, No.4, 481--502 (1998) (in English)
Mat. Sbornik 189, No.4, 3--24 (1998) (in Russian)

The dvi file for this article (in English) is available here

------ ------ ------

Approximation by entire functions
by A. Boivin and P.V. Paramonov

Abstract: Examples of the type of (new) results we can deduce from the theorems obtained in the paper "Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions" are given in the case when L is the Cauchy-Riemann or the Laplacian operator and the norm is the C^m norm.

Proceeding of a Conference in memory of A.F. Leont'ev, Nizhni Novgorod, Russia, June 2-5, 1997, 91-92.

The dvi file for this article is available here

------ ------ ------

Extension and Approximation of CR functions on tube manifolds
by A. Boivin and R. Dwilewicz

Abstract: In this paper we prove the following theorem: Any continuous CR function on a tube manifold t(N) = N x iRⁿ in Cⁿ, where N is an embedded submanifold of Rⁿ of class C², can be continuously CR extended to the tube over the convex hull of N. There are no additional assumptions on N, and in this sense, the theorem generalizes all known results of this type, including the classical Bochner tube theorem. Also it implies almost uniform approximation of CR functions on tubes by holomorphic polynomials in Cⁿ.

Trans. Amer. Math. Soc. 350, 1945-1956 (1998)

The dvi file for this article is available here

------ ------ ------

An example in tangential meromorphic approximation
by A. Boivin and A.H. Nersessian

Abstract: It is known that all sets of meromorphic uniform approximation in C which satisfy an additional condition involving the Gleason parts of the algebra R(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, though it is a set of tangential approximation, does not satisfy this extra condition on parts, and thus showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem.
© 1996 Academic Press Inc.

Journal of Approximation Theory 87, 103-111 (1996)

------ ------ ------

Meromorphic approximation in weighted L^p spaces
by A. Boivin, A. Bonilla and J.C. Fariña

Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995)
------ ------ ------
Uniform approximation by meromorphic functions on Riemann surfaces
by A. Boivin and B. Jiang

Abstract: A closed subset E of a Riemann surface S is called a set of uniform meromorphic approximation if every function f continuous on E and holomorphic on E⁰ can be approximated uniformly on E by meromorphic functions on S. We show that if E is a set of meromorphic approximation, then so is E \cap D for every compact parametric disk D. As a consequence, we obtain a generalization to Riemann surfaces of a well-known theorem of A.G. Vitushkin characterizing the compact sets of uniform meromorphic approximation. More precisely we prove that if K is a compact subset of an arbitrary Riemann surface then A(K) = H(K) if and only if for every point p in K, there exists a closed parametric disk D centred at p such that A(K \cap D) = H(K \cap D).

Journal d'Analyse Mathématique (to appear -- 19 pages)

The dvi file for this article is available here

------ ------ ------

The growth of an entire function and its Dirichlet coefficients and exponents.
by A. Boivin and C. Zhu

Abstract: Let F be an entire function represented by a (generalized) Dirichlet series F(s) = sum d_ne^{-t_n s} where the coefficients {d_n} and the exponents {t_n} are sequences of complex numbers. We introduce a modified (R)-order rho and modified (R)-type sigma and we obtain an estimate for |d_n| when n is sufficiently large in terms of rho, sigma and t_n. Other estimates relating rho and sigma to d_n and t_n are also obtained.

Complex Variables - Theory and Application, 48, 397-416 (2003)

The dvi file for this article is available here

------ ------ ------

On the completeness of the system { z ^t_n} in L².
by A. Boivin and C. Zhu

Abstract: Given an unbounded domain G located outside an angle domain with vertex at the origin, and a sequence of distinct complex numbers {t_n} satisfying n/|t_n| tends to D as n tends to infinity with D positive and finite, and |arg(t_n)| < A < pi/2, we obtain a completeness theorem for the system { z^t_n} in L²_a. The case with weight is also considered.
© 2002 Academic Press Inc.

Journal of Approximation Theory 118, 1-19 (2002)

------ ------ ------

Approximation on closed sets by analytic or meromorphic solutions
of elliptic equations and applications
by A. Boivin, P.M. Gauthier and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant complex coefficients and a class of functions (jet-distributions) which are defined on a (relatively) closed subset of a domain G in Rⁿ and which belong locally to a Banach space V, we consider the problem of approximating in the norm of V the functions in this class by ``analytic" and ``meromorphic" solutions of the equation Lu=0. We establish new Roth, Arakelyan (including tangential) and Carleman type theorems for a large class of Banach spaces V and operators L. Important applications to boundary value problems of solutions of homogeneous elliptic partial differential equations are obtained, including the solution of a generalized Dirichlet problem.

Canadian Mathematics Journal 54, 945-969 (2002)

------ ------ ------

On the L²-completeness of some systems of analytic functions
by A. Boivin and C. Zhu

Abstract: For an entire function f(z), a sequence of complex numbers {µ_n} and an unbounded domain G, the completeness of the system {f(µ_nz)} in L²_a(G) (mean-square approximation) is studied. A similar problem is discussed when f is analytic on the Riemann surface of the logarithm.
© 2001 OPA (Overseas Publishers Association) N.V.

Complex Variables - Theory and Application, 45, 273-295 (2001)

------ ------ ------

Holomorphic and harmonic approximation on Riemann surfaces
by A. Boivin and P.M. Gauthier

Abstract: The most important result on open Riemann surfaces is the analog of Runge's theorem on approximation by complex polynomials, due to Behnke and Stein. In this course, we begin by a discussion of Runge's theorems on polynomial and rational approximation on compact sets. This theory is refined and extended in various ways to Riemann surfaces. We also introduce a corresponding theory of harmonic approximation.
© 2001 Kluwer Academic Publishers

Approximation, Complex Analysis and Potential Theory
Proceedings of a NATO ASI held in Montreal, July 2000
N. Arakelian and P.M. Gauthier, eds.
NATO Science Series II, Vol. 37
Kluwer Academic Publishers 2001, 107--128

------ ------ ------

On radial limit functions for entire solutions of second order elliptic equations in R²
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L of order two with constant complex coefficients in R², we consider entire solutions of the equation Lu = 0 for which U(e^it) := lim u(re^it) exists for all t in [0, 2pi) as a finite limit in C as r goes to infinity. We characterize the possible ``radial limit functions" U. This is an analog of the work of Alice Roth for entire holomorphic functions. The results seem new even for harmonic functions.

Publicacions Matemàtiques, 42, No.2, 509-519 (1998)

The dvi file for this article is available here

------ ------ ------

Approximation by meromorphic and entire solutions
of elliptic equations in Banach spaces of distributions
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant coefficients and a class of functions (jets-distributions) which are defined on a closed, not necessarily compact, subset of Rⁿ and which belong locally to a Banach space V, we consider the approximation in the norm of V of the functions (jets) in this class by global (entire or meromorphic) solutions of the equation Lu=0. We establish Runge, Mergelyan, Roth and Arakelyan - type theorems for a large class of Banach spaces V and operators L, encompassing most of the previously considered generalizations of these theorems and obtaining new ones.

Sbornik Math. 189, No.4, 481--502 (1998) (in English)
Mat. Sbornik 189, No.4, 3--24 (1998) (in Russian)

The dvi file for this article (in English) is available here

------ ------ ------

Approximation by entire functions
by A. Boivin and P.V. Paramonov

Abstract: Examples of the type of (new) results we can deduce from the theorems obtained in the paper "Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions" are given in the case when L is the Cauchy-Riemann or the Laplacian operator and the norm is the C^m norm.

Proceeding of a Conference in memory of A.F. Leont'ev, Nizhni Novgorod, Russia, June 2-5, 1997, 91-92.

The dvi file for this article is available here

------ ------ ------

Extension and Approximation of CR functions on tube manifolds
by A. Boivin and R. Dwilewicz

Abstract: In this paper we prove the following theorem: Any continuous CR function on a tube manifold t(N) = N x iRⁿ in Cⁿ, where N is an embedded submanifold of Rⁿ of class C², can be continuously CR extended to the tube over the convex hull of N. There are no additional assumptions on N, and in this sense, the theorem generalizes all known results of this type, including the classical Bochner tube theorem. Also it implies almost uniform approximation of CR functions on tubes by holomorphic polynomials in Cⁿ.

Trans. Amer. Math. Soc. 350, 1945-1956 (1998)

The dvi file for this article is available here

------ ------ ------

An example in tangential meromorphic approximation
by A. Boivin and A.H. Nersessian

Abstract: It is known that all sets of meromorphic uniform approximation in C which satisfy an additional condition involving the Gleason parts of the algebra R(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, though it is a set of tangential approximation, does not satisfy this extra condition on parts, and thus showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem.
© 1996 Academic Press Inc.

Journal of Approximation Theory 87, 103-111 (1996)

------ ------ ------

Meromorphic approximation in weighted L^p spaces
by A. Boivin, A. Bonilla and J.C. Fariña

Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995): Abstract: A closed subset E of a Riemann surface S is called a set of uniform meromorphic approximation if every function f continuous on E and holomorphic on E⁰ can be approximated uniformly on E by meromorphic functions on S. We show that if E is a set of meromorphic approximation, then so is E \cap D for every compact parametric disk D. As a consequence, we obtain a generalization to Riemann surfaces of a well-known theorem of A.G. Vitushkin characterizing the compact sets of uniform meromorphic approximation. More precisely we prove that if K is a compact subset of an arbitrary Riemann surface then A(K) = H(K) if and only if for every point p in K, there exists a closed parametric disk D centred at p such that A(K \cap D) = H(K \cap D).

Journal d'Analyse Mathématique (to appear -- 19 pages)

The dvi file for this article is available here

------ ------ ------

The growth of an entire function and its Dirichlet coefficients and exponents.
by A. Boivin and C. Zhu

Abstract: Let F be an entire function represented by a (generalized) Dirichlet series F(s) = sum d_ne^{-t_n s} where the coefficients {d_n} and the exponents {t_n} are sequences of complex numbers. We introduce a modified (R)-order rho and modified (R)-type sigma and we obtain an estimate for |d_n| when n is sufficiently large in terms of rho, sigma and t_n. Other estimates relating rho and sigma to d_n and t_n are also obtained.

Complex Variables - Theory and Application, 48, 397-416 (2003)

The dvi file for this article is available here

------ ------ ------

On the completeness of the system { z ^t_n} in L².
by A. Boivin and C. Zhu

Abstract: Given an unbounded domain G located outside an angle domain with vertex at the origin, and a sequence of distinct complex numbers {t_n} satisfying n/|t_n| tends to D as n tends to infinity with D positive and finite, and |arg(t_n)| < A < pi/2, we obtain a completeness theorem for the system { z^t_n} in L²_a. The case with weight is also considered.
© 2002 Academic Press Inc.

Journal of Approximation Theory 118, 1-19 (2002)

------ ------ ------

Approximation on closed sets by analytic or meromorphic solutions
of elliptic equations and applications
by A. Boivin, P.M. Gauthier and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant complex coefficients and a class of functions (jet-distributions) which are defined on a (relatively) closed subset of a domain G in Rⁿ and which belong locally to a Banach space V, we consider the problem of approximating in the norm of V the functions in this class by ``analytic" and ``meromorphic" solutions of the equation Lu=0. We establish new Roth, Arakelyan (including tangential) and Carleman type theorems for a large class of Banach spaces V and operators L. Important applications to boundary value problems of solutions of homogeneous elliptic partial differential equations are obtained, including the solution of a generalized Dirichlet problem.

Canadian Mathematics Journal 54, 945-969 (2002)

------ ------ ------

On the L²-completeness of some systems of analytic functions
by A. Boivin and C. Zhu

Abstract: For an entire function f(z), a sequence of complex numbers {µ_n} and an unbounded domain G, the completeness of the system {f(µ_nz)} in L²_a(G) (mean-square approximation) is studied. A similar problem is discussed when f is analytic on the Riemann surface of the logarithm.
© 2001 OPA (Overseas Publishers Association) N.V.

Complex Variables - Theory and Application, 45, 273-295 (2001)

------ ------ ------

Holomorphic and harmonic approximation on Riemann surfaces
by A. Boivin and P.M. Gauthier

Abstract: The most important result on open Riemann surfaces is the analog of Runge's theorem on approximation by complex polynomials, due to Behnke and Stein. In this course, we begin by a discussion of Runge's theorems on polynomial and rational approximation on compact sets. This theory is refined and extended in various ways to Riemann surfaces. We also introduce a corresponding theory of harmonic approximation.
© 2001 Kluwer Academic Publishers

Approximation, Complex Analysis and Potential Theory
Proceedings of a NATO ASI held in Montreal, July 2000
N. Arakelian and P.M. Gauthier, eds.
NATO Science Series II, Vol. 37
Kluwer Academic Publishers 2001, 107--128

------ ------ ------

On radial limit functions for entire solutions of second order elliptic equations in R²
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L of order two with constant complex coefficients in R², we consider entire solutions of the equation Lu = 0 for which U(e^it) := lim u(re^it) exists for all t in [0, 2pi) as a finite limit in C as r goes to infinity. We characterize the possible ``radial limit functions" U. This is an analog of the work of Alice Roth for entire holomorphic functions. The results seem new even for harmonic functions.

Publicacions Matemàtiques, 42, No.2, 509-519 (1998)

The dvi file for this article is available here

------ ------ ------

Approximation by meromorphic and entire solutions
of elliptic equations in Banach spaces of distributions
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant coefficients and a class of functions (jets-distributions) which are defined on a closed, not necessarily compact, subset of Rⁿ and which belong locally to a Banach space V, we consider the approximation in the norm of V of the functions (jets) in this class by global (entire or meromorphic) solutions of the equation Lu=0. We establish Runge, Mergelyan, Roth and Arakelyan - type theorems for a large class of Banach spaces V and operators L, encompassing most of the previously considered generalizations of these theorems and obtaining new ones.

Sbornik Math. 189, No.4, 481--502 (1998) (in English)
Mat. Sbornik 189, No.4, 3--24 (1998) (in Russian)

The dvi file for this article (in English) is available here

------ ------ ------

Approximation by entire functions
by A. Boivin and P.V. Paramonov

Abstract: Examples of the type of (new) results we can deduce from the theorems obtained in the paper "Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions" are given in the case when L is the Cauchy-Riemann or the Laplacian operator and the norm is the C^m norm.

Proceeding of a Conference in memory of A.F. Leont'ev, Nizhni Novgorod, Russia, June 2-5, 1997, 91-92.

The dvi file for this article is available here

------ ------ ------

Extension and Approximation of CR functions on tube manifolds
by A. Boivin and R. Dwilewicz

Abstract: In this paper we prove the following theorem: Any continuous CR function on a tube manifold t(N) = N x iRⁿ in Cⁿ, where N is an embedded submanifold of Rⁿ of class C², can be continuously CR extended to the tube over the convex hull of N. There are no additional assumptions on N, and in this sense, the theorem generalizes all known results of this type, including the classical Bochner tube theorem. Also it implies almost uniform approximation of CR functions on tubes by holomorphic polynomials in Cⁿ.

Trans. Amer. Math. Soc. 350, 1945-1956 (1998)

The dvi file for this article is available here

------ ------ ------

An example in tangential meromorphic approximation
by A. Boivin and A.H. Nersessian

Abstract: It is known that all sets of meromorphic uniform approximation in C which satisfy an additional condition involving the Gleason parts of the algebra R(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, though it is a set of tangential approximation, does not satisfy this extra condition on parts, and thus showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem.
© 1996 Academic Press Inc.

Journal of Approximation Theory 87, 103-111 (1996)

------ ------ ------

Meromorphic approximation in weighted L^p spaces
by A. Boivin, A. Bonilla and J.C. Fariña

Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995)
------ ------ ------

The growth of an entire function and its Dirichlet coefficients and exponents.
by A. Boivin and C. Zhu

Abstract: Let F be an entire function represented by a (generalized) Dirichlet series F(s) = sum d_ne^{-t_n s} where the coefficients {d_n} and the exponents {t_n} are sequences of complex numbers. We introduce a modified (R)-order rho and modified (R)-type sigma and we obtain an estimate for |d_n| when n is sufficiently large in terms of rho, sigma and t_n. Other estimates relating rho and sigma to d_n and t_n are also obtained.

Complex Variables - Theory and Application, 48, 397-416 (2003)

The dvi file for this article is available here

------ ------ ------

On the completeness of the system { z ^t_n} in L².
by A. Boivin and C. Zhu

Abstract: Given an unbounded domain G located outside an angle domain with vertex at the origin, and a sequence of distinct complex numbers {t_n} satisfying n/|t_n| tends to D as n tends to infinity with D positive and finite, and |arg(t_n)| < A < pi/2, we obtain a completeness theorem for the system { z^t_n} in L²_a. The case with weight is also considered.
© 2002 Academic Press Inc.

Journal of Approximation Theory 118, 1-19 (2002)

------ ------ ------

Approximation on closed sets by analytic or meromorphic solutions
of elliptic equations and applications
by A. Boivin, P.M. Gauthier and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant complex coefficients and a class of functions (jet-distributions) which are defined on a (relatively) closed subset of a domain G in Rⁿ and which belong locally to a Banach space V, we consider the problem of approximating in the norm of V the functions in this class by ``analytic" and ``meromorphic" solutions of the equation Lu=0. We establish new Roth, Arakelyan (including tangential) and Carleman type theorems for a large class of Banach spaces V and operators L. Important applications to boundary value problems of solutions of homogeneous elliptic partial differential equations are obtained, including the solution of a generalized Dirichlet problem.

Canadian Mathematics Journal 54, 945-969 (2002)

------ ------ ------

On the L²-completeness of some systems of analytic functions
by A. Boivin and C. Zhu

Abstract: For an entire function f(z), a sequence of complex numbers {µ_n} and an unbounded domain G, the completeness of the system {f(µ_nz)} in L²_a(G) (mean-square approximation) is studied. A similar problem is discussed when f is analytic on the Riemann surface of the logarithm.
© 2001 OPA (Overseas Publishers Association) N.V.

Complex Variables - Theory and Application, 45, 273-295 (2001)

------ ------ ------

Holomorphic and harmonic approximation on Riemann surfaces
by A. Boivin and P.M. Gauthier

Abstract: The most important result on open Riemann surfaces is the analog of Runge's theorem on approximation by complex polynomials, due to Behnke and Stein. In this course, we begin by a discussion of Runge's theorems on polynomial and rational approximation on compact sets. This theory is refined and extended in various ways to Riemann surfaces. We also introduce a corresponding theory of harmonic approximation.
© 2001 Kluwer Academic Publishers

Approximation, Complex Analysis and Potential Theory
Proceedings of a NATO ASI held in Montreal, July 2000
N. Arakelian and P.M. Gauthier, eds.
NATO Science Series II, Vol. 37
Kluwer Academic Publishers 2001, 107--128

------ ------ ------

On radial limit functions for entire solutions of second order elliptic equations in R²
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L of order two with constant complex coefficients in R², we consider entire solutions of the equation Lu = 0 for which U(e^it) := lim u(re^it) exists for all t in [0, 2pi) as a finite limit in C as r goes to infinity. We characterize the possible ``radial limit functions" U. This is an analog of the work of Alice Roth for entire holomorphic functions. The results seem new even for harmonic functions.

Publicacions Matemàtiques, 42, No.2, 509-519 (1998)

The dvi file for this article is available here

------ ------ ------

Approximation by meromorphic and entire solutions
of elliptic equations in Banach spaces of distributions
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant coefficients and a class of functions (jets-distributions) which are defined on a closed, not necessarily compact, subset of Rⁿ and which belong locally to a Banach space V, we consider the approximation in the norm of V of the functions (jets) in this class by global (entire or meromorphic) solutions of the equation Lu=0. We establish Runge, Mergelyan, Roth and Arakelyan - type theorems for a large class of Banach spaces V and operators L, encompassing most of the previously considered generalizations of these theorems and obtaining new ones.

Sbornik Math. 189, No.4, 481--502 (1998) (in English)
Mat. Sbornik 189, No.4, 3--24 (1998) (in Russian)

The dvi file for this article (in English) is available here

------ ------ ------

Approximation by entire functions
by A. Boivin and P.V. Paramonov

Abstract: Examples of the type of (new) results we can deduce from the theorems obtained in the paper "Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions" are given in the case when L is the Cauchy-Riemann or the Laplacian operator and the norm is the C^m norm.

Proceeding of a Conference in memory of A.F. Leont'ev, Nizhni Novgorod, Russia, June 2-5, 1997, 91-92.

The dvi file for this article is available here

------ ------ ------

Extension and Approximation of CR functions on tube manifolds
by A. Boivin and R. Dwilewicz

Abstract: In this paper we prove the following theorem: Any continuous CR function on a tube manifold t(N) = N x iRⁿ in Cⁿ, where N is an embedded submanifold of Rⁿ of class C², can be continuously CR extended to the tube over the convex hull of N. There are no additional assumptions on N, and in this sense, the theorem generalizes all known results of this type, including the classical Bochner tube theorem. Also it implies almost uniform approximation of CR functions on tubes by holomorphic polynomials in Cⁿ.

Trans. Amer. Math. Soc. 350, 1945-1956 (1998)

The dvi file for this article is available here

------ ------ ------

An example in tangential meromorphic approximation
by A. Boivin and A.H. Nersessian

Abstract: It is known that all sets of meromorphic uniform approximation in C which satisfy an additional condition involving the Gleason parts of the algebra R(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, though it is a set of tangential approximation, does not satisfy this extra condition on parts, and thus showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem.
© 1996 Academic Press Inc.

Journal of Approximation Theory 87, 103-111 (1996)

------ ------ ------

Meromorphic approximation in weighted L^p spaces
by A. Boivin, A. Bonilla and J.C. Fariña

Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995): Abstract: Let F be an entire function represented by a (generalized) Dirichlet series F(s) = sum d_ne^{-t_n s} where the coefficients {d_n} and the exponents {t_n} are sequences of complex numbers. We introduce a modified (R)-order rho and modified (R)-type sigma and we obtain an estimate for |d_n| when n is sufficiently large in terms of rho, sigma and t_n. Other estimates relating rho and sigma to d_n and t_n are also obtained.

Complex Variables - Theory and Application, 48, 397-416 (2003)

The dvi file for this article is available here

------ ------ ------

On the completeness of the system { z ^t_n} in L².
by A. Boivin and C. Zhu

Abstract: Given an unbounded domain G located outside an angle domain with vertex at the origin, and a sequence of distinct complex numbers {t_n} satisfying n/|t_n| tends to D as n tends to infinity with D positive and finite, and |arg(t_n)| < A < pi/2, we obtain a completeness theorem for the system { z^t_n} in L²_a. The case with weight is also considered.
© 2002 Academic Press Inc.

Journal of Approximation Theory 118, 1-19 (2002)

------ ------ ------

Approximation on closed sets by analytic or meromorphic solutions
of elliptic equations and applications
by A. Boivin, P.M. Gauthier and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant complex coefficients and a class of functions (jet-distributions) which are defined on a (relatively) closed subset of a domain G in Rⁿ and which belong locally to a Banach space V, we consider the problem of approximating in the norm of V the functions in this class by ``analytic" and ``meromorphic" solutions of the equation Lu=0. We establish new Roth, Arakelyan (including tangential) and Carleman type theorems for a large class of Banach spaces V and operators L. Important applications to boundary value problems of solutions of homogeneous elliptic partial differential equations are obtained, including the solution of a generalized Dirichlet problem.

Canadian Mathematics Journal 54, 945-969 (2002)

------ ------ ------

On the L²-completeness of some systems of analytic functions
by A. Boivin and C. Zhu

Abstract: For an entire function f(z), a sequence of complex numbers {µ_n} and an unbounded domain G, the completeness of the system {f(µ_nz)} in L²_a(G) (mean-square approximation) is studied. A similar problem is discussed when f is analytic on the Riemann surface of the logarithm.
© 2001 OPA (Overseas Publishers Association) N.V.

Complex Variables - Theory and Application, 45, 273-295 (2001)

------ ------ ------

Holomorphic and harmonic approximation on Riemann surfaces
by A. Boivin and P.M. Gauthier

Abstract: The most important result on open Riemann surfaces is the analog of Runge's theorem on approximation by complex polynomials, due to Behnke and Stein. In this course, we begin by a discussion of Runge's theorems on polynomial and rational approximation on compact sets. This theory is refined and extended in various ways to Riemann surfaces. We also introduce a corresponding theory of harmonic approximation.
© 2001 Kluwer Academic Publishers

Approximation, Complex Analysis and Potential Theory
Proceedings of a NATO ASI held in Montreal, July 2000
N. Arakelian and P.M. Gauthier, eds.
NATO Science Series II, Vol. 37
Kluwer Academic Publishers 2001, 107--128

------ ------ ------

On radial limit functions for entire solutions of second order elliptic equations in R²
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L of order two with constant complex coefficients in R², we consider entire solutions of the equation Lu = 0 for which U(e^it) := lim u(re^it) exists for all t in [0, 2pi) as a finite limit in C as r goes to infinity. We characterize the possible ``radial limit functions" U. This is an analog of the work of Alice Roth for entire holomorphic functions. The results seem new even for harmonic functions.

Publicacions Matemàtiques, 42, No.2, 509-519 (1998)

The dvi file for this article is available here

------ ------ ------

Approximation by meromorphic and entire solutions
of elliptic equations in Banach spaces of distributions
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant coefficients and a class of functions (jets-distributions) which are defined on a closed, not necessarily compact, subset of Rⁿ and which belong locally to a Banach space V, we consider the approximation in the norm of V of the functions (jets) in this class by global (entire or meromorphic) solutions of the equation Lu=0. We establish Runge, Mergelyan, Roth and Arakelyan - type theorems for a large class of Banach spaces V and operators L, encompassing most of the previously considered generalizations of these theorems and obtaining new ones.

Sbornik Math. 189, No.4, 481--502 (1998) (in English)
Mat. Sbornik 189, No.4, 3--24 (1998) (in Russian)

The dvi file for this article (in English) is available here

------ ------ ------

Approximation by entire functions
by A. Boivin and P.V. Paramonov

Abstract: Examples of the type of (new) results we can deduce from the theorems obtained in the paper "Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions" are given in the case when L is the Cauchy-Riemann or the Laplacian operator and the norm is the C^m norm.

Proceeding of a Conference in memory of A.F. Leont'ev, Nizhni Novgorod, Russia, June 2-5, 1997, 91-92.

The dvi file for this article is available here

------ ------ ------

Extension and Approximation of CR functions on tube manifolds
by A. Boivin and R. Dwilewicz

Abstract: In this paper we prove the following theorem: Any continuous CR function on a tube manifold t(N) = N x iRⁿ in Cⁿ, where N is an embedded submanifold of Rⁿ of class C², can be continuously CR extended to the tube over the convex hull of N. There are no additional assumptions on N, and in this sense, the theorem generalizes all known results of this type, including the classical Bochner tube theorem. Also it implies almost uniform approximation of CR functions on tubes by holomorphic polynomials in Cⁿ.

Trans. Amer. Math. Soc. 350, 1945-1956 (1998)

The dvi file for this article is available here

------ ------ ------

An example in tangential meromorphic approximation
by A. Boivin and A.H. Nersessian

Abstract: It is known that all sets of meromorphic uniform approximation in C which satisfy an additional condition involving the Gleason parts of the algebra R(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, though it is a set of tangential approximation, does not satisfy this extra condition on parts, and thus showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem.
© 1996 Academic Press Inc.

Journal of Approximation Theory 87, 103-111 (1996)

------ ------ ------

Meromorphic approximation in weighted L^p spaces
by A. Boivin, A. Bonilla and J.C. Fariña

Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995)
------ ------ ------

On the completeness of the system { z ^t_n} in L².
by A. Boivin and C. Zhu

Abstract: Given an unbounded domain G located outside an angle domain with vertex at the origin, and a sequence of distinct complex numbers {t_n} satisfying n/|t_n| tends to D as n tends to infinity with D positive and finite, and |arg(t_n)| < A < pi/2, we obtain a completeness theorem for the system { z^t_n} in L²_a. The case with weight is also considered.
© 2002 Academic Press Inc.

Journal of Approximation Theory 118, 1-19 (2002)

------ ------ ------

Approximation on closed sets by analytic or meromorphic solutions
of elliptic equations and applications
by A. Boivin, P.M. Gauthier and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant complex coefficients and a class of functions (jet-distributions) which are defined on a (relatively) closed subset of a domain G in Rⁿ and which belong locally to a Banach space V, we consider the problem of approximating in the norm of V the functions in this class by ``analytic" and ``meromorphic" solutions of the equation Lu=0. We establish new Roth, Arakelyan (including tangential) and Carleman type theorems for a large class of Banach spaces V and operators L. Important applications to boundary value problems of solutions of homogeneous elliptic partial differential equations are obtained, including the solution of a generalized Dirichlet problem.

Canadian Mathematics Journal 54, 945-969 (2002)

------ ------ ------

On the L²-completeness of some systems of analytic functions
by A. Boivin and C. Zhu

Abstract: For an entire function f(z), a sequence of complex numbers {µ_n} and an unbounded domain G, the completeness of the system {f(µ_nz)} in L²_a(G) (mean-square approximation) is studied. A similar problem is discussed when f is analytic on the Riemann surface of the logarithm.
© 2001 OPA (Overseas Publishers Association) N.V.

Complex Variables - Theory and Application, 45, 273-295 (2001)

------ ------ ------

Holomorphic and harmonic approximation on Riemann surfaces
by A. Boivin and P.M. Gauthier

Abstract: The most important result on open Riemann surfaces is the analog of Runge's theorem on approximation by complex polynomials, due to Behnke and Stein. In this course, we begin by a discussion of Runge's theorems on polynomial and rational approximation on compact sets. This theory is refined and extended in various ways to Riemann surfaces. We also introduce a corresponding theory of harmonic approximation.
© 2001 Kluwer Academic Publishers

Approximation, Complex Analysis and Potential Theory
Proceedings of a NATO ASI held in Montreal, July 2000
N. Arakelian and P.M. Gauthier, eds.
NATO Science Series II, Vol. 37
Kluwer Academic Publishers 2001, 107--128

------ ------ ------

On radial limit functions for entire solutions of second order elliptic equations in R²
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L of order two with constant complex coefficients in R², we consider entire solutions of the equation Lu = 0 for which U(e^it) := lim u(re^it) exists for all t in [0, 2pi) as a finite limit in C as r goes to infinity. We characterize the possible ``radial limit functions" U. This is an analog of the work of Alice Roth for entire holomorphic functions. The results seem new even for harmonic functions.

Publicacions Matemàtiques, 42, No.2, 509-519 (1998)

The dvi file for this article is available here

------ ------ ------

Approximation by meromorphic and entire solutions
of elliptic equations in Banach spaces of distributions
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant coefficients and a class of functions (jets-distributions) which are defined on a closed, not necessarily compact, subset of Rⁿ and which belong locally to a Banach space V, we consider the approximation in the norm of V of the functions (jets) in this class by global (entire or meromorphic) solutions of the equation Lu=0. We establish Runge, Mergelyan, Roth and Arakelyan - type theorems for a large class of Banach spaces V and operators L, encompassing most of the previously considered generalizations of these theorems and obtaining new ones.

Sbornik Math. 189, No.4, 481--502 (1998) (in English)
Mat. Sbornik 189, No.4, 3--24 (1998) (in Russian)

The dvi file for this article (in English) is available here

------ ------ ------

Approximation by entire functions
by A. Boivin and P.V. Paramonov

Abstract: Examples of the type of (new) results we can deduce from the theorems obtained in the paper "Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions" are given in the case when L is the Cauchy-Riemann or the Laplacian operator and the norm is the C^m norm.

Proceeding of a Conference in memory of A.F. Leont'ev, Nizhni Novgorod, Russia, June 2-5, 1997, 91-92.

The dvi file for this article is available here

------ ------ ------

Extension and Approximation of CR functions on tube manifolds
by A. Boivin and R. Dwilewicz

Abstract: In this paper we prove the following theorem: Any continuous CR function on a tube manifold t(N) = N x iRⁿ in Cⁿ, where N is an embedded submanifold of Rⁿ of class C², can be continuously CR extended to the tube over the convex hull of N. There are no additional assumptions on N, and in this sense, the theorem generalizes all known results of this type, including the classical Bochner tube theorem. Also it implies almost uniform approximation of CR functions on tubes by holomorphic polynomials in Cⁿ.

Trans. Amer. Math. Soc. 350, 1945-1956 (1998)

The dvi file for this article is available here

------ ------ ------

An example in tangential meromorphic approximation
by A. Boivin and A.H. Nersessian

Abstract: It is known that all sets of meromorphic uniform approximation in C which satisfy an additional condition involving the Gleason parts of the algebra R(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, though it is a set of tangential approximation, does not satisfy this extra condition on parts, and thus showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem.
© 1996 Academic Press Inc.

Journal of Approximation Theory 87, 103-111 (1996)

------ ------ ------

Meromorphic approximation in weighted L^p spaces
by A. Boivin, A. Bonilla and J.C. Fariña

Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995): Abstract: Given an unbounded domain G located outside an angle domain with vertex at the origin, and a sequence of distinct complex numbers {t_n} satisfying n/|t_n| tends to D as n tends to infinity with D positive and finite, and |arg(t_n)| < A < pi/2, we obtain a completeness theorem for the system { z^t_n} in L²_a. The case with weight is also considered.
© 2002 Academic Press Inc.

Journal of Approximation Theory 118, 1-19 (2002)

------ ------ ------

Approximation on closed sets by analytic or meromorphic solutions
of elliptic equations and applications
by A. Boivin, P.M. Gauthier and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant complex coefficients and a class of functions (jet-distributions) which are defined on a (relatively) closed subset of a domain G in Rⁿ and which belong locally to a Banach space V, we consider the problem of approximating in the norm of V the functions in this class by ``analytic" and ``meromorphic" solutions of the equation Lu=0. We establish new Roth, Arakelyan (including tangential) and Carleman type theorems for a large class of Banach spaces V and operators L. Important applications to boundary value problems of solutions of homogeneous elliptic partial differential equations are obtained, including the solution of a generalized Dirichlet problem.

Canadian Mathematics Journal 54, 945-969 (2002)

------ ------ ------

On the L²-completeness of some systems of analytic functions
by A. Boivin and C. Zhu

Abstract: For an entire function f(z), a sequence of complex numbers {µ_n} and an unbounded domain G, the completeness of the system {f(µ_nz)} in L²_a(G) (mean-square approximation) is studied. A similar problem is discussed when f is analytic on the Riemann surface of the logarithm.
© 2001 OPA (Overseas Publishers Association) N.V.

Complex Variables - Theory and Application, 45, 273-295 (2001)

------ ------ ------

Holomorphic and harmonic approximation on Riemann surfaces
by A. Boivin and P.M. Gauthier

Abstract: The most important result on open Riemann surfaces is the analog of Runge's theorem on approximation by complex polynomials, due to Behnke and Stein. In this course, we begin by a discussion of Runge's theorems on polynomial and rational approximation on compact sets. This theory is refined and extended in various ways to Riemann surfaces. We also introduce a corresponding theory of harmonic approximation.
© 2001 Kluwer Academic Publishers

Approximation, Complex Analysis and Potential Theory
Proceedings of a NATO ASI held in Montreal, July 2000
N. Arakelian and P.M. Gauthier, eds.
NATO Science Series II, Vol. 37
Kluwer Academic Publishers 2001, 107--128

------ ------ ------

On radial limit functions for entire solutions of second order elliptic equations in R²
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L of order two with constant complex coefficients in R², we consider entire solutions of the equation Lu = 0 for which U(e^it) := lim u(re^it) exists for all t in [0, 2pi) as a finite limit in C as r goes to infinity. We characterize the possible ``radial limit functions" U. This is an analog of the work of Alice Roth for entire holomorphic functions. The results seem new even for harmonic functions.

Publicacions Matemàtiques, 42, No.2, 509-519 (1998)

The dvi file for this article is available here

------ ------ ------

Approximation by meromorphic and entire solutions
of elliptic equations in Banach spaces of distributions
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant coefficients and a class of functions (jets-distributions) which are defined on a closed, not necessarily compact, subset of Rⁿ and which belong locally to a Banach space V, we consider the approximation in the norm of V of the functions (jets) in this class by global (entire or meromorphic) solutions of the equation Lu=0. We establish Runge, Mergelyan, Roth and Arakelyan - type theorems for a large class of Banach spaces V and operators L, encompassing most of the previously considered generalizations of these theorems and obtaining new ones.

Sbornik Math. 189, No.4, 481--502 (1998) (in English)
Mat. Sbornik 189, No.4, 3--24 (1998) (in Russian)

The dvi file for this article (in English) is available here

------ ------ ------

Approximation by entire functions
by A. Boivin and P.V. Paramonov

Abstract: Examples of the type of (new) results we can deduce from the theorems obtained in the paper "Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions" are given in the case when L is the Cauchy-Riemann or the Laplacian operator and the norm is the C^m norm.

Proceeding of a Conference in memory of A.F. Leont'ev, Nizhni Novgorod, Russia, June 2-5, 1997, 91-92.

The dvi file for this article is available here

------ ------ ------

Extension and Approximation of CR functions on tube manifolds
by A. Boivin and R. Dwilewicz

Abstract: In this paper we prove the following theorem: Any continuous CR function on a tube manifold t(N) = N x iRⁿ in Cⁿ, where N is an embedded submanifold of Rⁿ of class C², can be continuously CR extended to the tube over the convex hull of N. There are no additional assumptions on N, and in this sense, the theorem generalizes all known results of this type, including the classical Bochner tube theorem. Also it implies almost uniform approximation of CR functions on tubes by holomorphic polynomials in Cⁿ.

Trans. Amer. Math. Soc. 350, 1945-1956 (1998)

The dvi file for this article is available here

------ ------ ------

An example in tangential meromorphic approximation
by A. Boivin and A.H. Nersessian

Abstract: It is known that all sets of meromorphic uniform approximation in C which satisfy an additional condition involving the Gleason parts of the algebra R(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, though it is a set of tangential approximation, does not satisfy this extra condition on parts, and thus showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem.
© 1996 Academic Press Inc.

Journal of Approximation Theory 87, 103-111 (1996)

------ ------ ------

Meromorphic approximation in weighted L^p spaces
by A. Boivin, A. Bonilla and J.C. Fariña

Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995)
------ ------ ------

Approximation on closed sets by analytic or meromorphic solutions
of elliptic equations and applications
by A. Boivin, P.M. Gauthier and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant complex coefficients and a class of functions (jet-distributions) which are defined on a (relatively) closed subset of a domain G in Rⁿ and which belong locally to a Banach space V, we consider the problem of approximating in the norm of V the functions in this class by ``analytic" and ``meromorphic" solutions of the equation Lu=0. We establish new Roth, Arakelyan (including tangential) and Carleman type theorems for a large class of Banach spaces V and operators L. Important applications to boundary value problems of solutions of homogeneous elliptic partial differential equations are obtained, including the solution of a generalized Dirichlet problem.

Canadian Mathematics Journal 54, 945-969 (2002)

------ ------ ------

On the L²-completeness of some systems of analytic functions
by A. Boivin and C. Zhu

Abstract: For an entire function f(z), a sequence of complex numbers {µ_n} and an unbounded domain G, the completeness of the system {f(µ_nz)} in L²_a(G) (mean-square approximation) is studied. A similar problem is discussed when f is analytic on the Riemann surface of the logarithm.
© 2001 OPA (Overseas Publishers Association) N.V.

Complex Variables - Theory and Application, 45, 273-295 (2001)

------ ------ ------

Holomorphic and harmonic approximation on Riemann surfaces
by A. Boivin and P.M. Gauthier

Abstract: The most important result on open Riemann surfaces is the analog of Runge's theorem on approximation by complex polynomials, due to Behnke and Stein. In this course, we begin by a discussion of Runge's theorems on polynomial and rational approximation on compact sets. This theory is refined and extended in various ways to Riemann surfaces. We also introduce a corresponding theory of harmonic approximation.
© 2001 Kluwer Academic Publishers

Approximation, Complex Analysis and Potential Theory
Proceedings of a NATO ASI held in Montreal, July 2000
N. Arakelian and P.M. Gauthier, eds.
NATO Science Series II, Vol. 37
Kluwer Academic Publishers 2001, 107--128

------ ------ ------

On radial limit functions for entire solutions of second order elliptic equations in R²
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L of order two with constant complex coefficients in R², we consider entire solutions of the equation Lu = 0 for which U(e^it) := lim u(re^it) exists for all t in [0, 2pi) as a finite limit in C as r goes to infinity. We characterize the possible ``radial limit functions" U. This is an analog of the work of Alice Roth for entire holomorphic functions. The results seem new even for harmonic functions.

Publicacions Matemàtiques, 42, No.2, 509-519 (1998)

The dvi file for this article is available here

------ ------ ------

Approximation by meromorphic and entire solutions
of elliptic equations in Banach spaces of distributions
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant coefficients and a class of functions (jets-distributions) which are defined on a closed, not necessarily compact, subset of Rⁿ and which belong locally to a Banach space V, we consider the approximation in the norm of V of the functions (jets) in this class by global (entire or meromorphic) solutions of the equation Lu=0. We establish Runge, Mergelyan, Roth and Arakelyan - type theorems for a large class of Banach spaces V and operators L, encompassing most of the previously considered generalizations of these theorems and obtaining new ones.

Sbornik Math. 189, No.4, 481--502 (1998) (in English)
Mat. Sbornik 189, No.4, 3--24 (1998) (in Russian)

The dvi file for this article (in English) is available here

------ ------ ------

Approximation by entire functions
by A. Boivin and P.V. Paramonov

Abstract: Examples of the type of (new) results we can deduce from the theorems obtained in the paper "Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions" are given in the case when L is the Cauchy-Riemann or the Laplacian operator and the norm is the C^m norm.

Proceeding of a Conference in memory of A.F. Leont'ev, Nizhni Novgorod, Russia, June 2-5, 1997, 91-92.

The dvi file for this article is available here

------ ------ ------

Extension and Approximation of CR functions on tube manifolds
by A. Boivin and R. Dwilewicz

Abstract: In this paper we prove the following theorem: Any continuous CR function on a tube manifold t(N) = N x iRⁿ in Cⁿ, where N is an embedded submanifold of Rⁿ of class C², can be continuously CR extended to the tube over the convex hull of N. There are no additional assumptions on N, and in this sense, the theorem generalizes all known results of this type, including the classical Bochner tube theorem. Also it implies almost uniform approximation of CR functions on tubes by holomorphic polynomials in Cⁿ.

Trans. Amer. Math. Soc. 350, 1945-1956 (1998)

The dvi file for this article is available here

------ ------ ------

An example in tangential meromorphic approximation
by A. Boivin and A.H. Nersessian

Abstract: It is known that all sets of meromorphic uniform approximation in C which satisfy an additional condition involving the Gleason parts of the algebra R(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, though it is a set of tangential approximation, does not satisfy this extra condition on parts, and thus showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem.
© 1996 Academic Press Inc.

Journal of Approximation Theory 87, 103-111 (1996)

------ ------ ------

Meromorphic approximation in weighted L^p spaces
by A. Boivin, A. Bonilla and J.C. Fariña

Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995): Abstract: Given a homogeneous elliptic partial differential operator L with constant complex coefficients and a class of functions (jet-distributions) which are defined on a (relatively) closed subset of a domain G in Rⁿ and which belong locally to a Banach space V, we consider the problem of approximating in the norm of V the functions in this class by ``analytic" and ``meromorphic" solutions of the equation Lu=0. We establish new Roth, Arakelyan (including tangential) and Carleman type theorems for a large class of Banach spaces V and operators L. Important applications to boundary value problems of solutions of homogeneous elliptic partial differential equations are obtained, including the solution of a generalized Dirichlet problem.

Canadian Mathematics Journal 54, 945-969 (2002)

------ ------ ------

On the L²-completeness of some systems of analytic functions
by A. Boivin and C. Zhu

Abstract: For an entire function f(z), a sequence of complex numbers {µ_n} and an unbounded domain G, the completeness of the system {f(µ_nz)} in L²_a(G) (mean-square approximation) is studied. A similar problem is discussed when f is analytic on the Riemann surface of the logarithm.
© 2001 OPA (Overseas Publishers Association) N.V.

Complex Variables - Theory and Application, 45, 273-295 (2001)

------ ------ ------

Holomorphic and harmonic approximation on Riemann surfaces
by A. Boivin and P.M. Gauthier

Abstract: The most important result on open Riemann surfaces is the analog of Runge's theorem on approximation by complex polynomials, due to Behnke and Stein. In this course, we begin by a discussion of Runge's theorems on polynomial and rational approximation on compact sets. This theory is refined and extended in various ways to Riemann surfaces. We also introduce a corresponding theory of harmonic approximation.
© 2001 Kluwer Academic Publishers

Approximation, Complex Analysis and Potential Theory
Proceedings of a NATO ASI held in Montreal, July 2000
N. Arakelian and P.M. Gauthier, eds.
NATO Science Series II, Vol. 37
Kluwer Academic Publishers 2001, 107--128

------ ------ ------

On radial limit functions for entire solutions of second order elliptic equations in R²
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L of order two with constant complex coefficients in R², we consider entire solutions of the equation Lu = 0 for which U(e^it) := lim u(re^it) exists for all t in [0, 2pi) as a finite limit in C as r goes to infinity. We characterize the possible ``radial limit functions" U. This is an analog of the work of Alice Roth for entire holomorphic functions. The results seem new even for harmonic functions.

Publicacions Matemàtiques, 42, No.2, 509-519 (1998)

The dvi file for this article is available here

------ ------ ------

Approximation by meromorphic and entire solutions
of elliptic equations in Banach spaces of distributions
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant coefficients and a class of functions (jets-distributions) which are defined on a closed, not necessarily compact, subset of Rⁿ and which belong locally to a Banach space V, we consider the approximation in the norm of V of the functions (jets) in this class by global (entire or meromorphic) solutions of the equation Lu=0. We establish Runge, Mergelyan, Roth and Arakelyan - type theorems for a large class of Banach spaces V and operators L, encompassing most of the previously considered generalizations of these theorems and obtaining new ones.

Sbornik Math. 189, No.4, 481--502 (1998) (in English)
Mat. Sbornik 189, No.4, 3--24 (1998) (in Russian)

The dvi file for this article (in English) is available here

------ ------ ------

Approximation by entire functions
by A. Boivin and P.V. Paramonov

Abstract: Examples of the type of (new) results we can deduce from the theorems obtained in the paper "Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions" are given in the case when L is the Cauchy-Riemann or the Laplacian operator and the norm is the C^m norm.

Proceeding of a Conference in memory of A.F. Leont'ev, Nizhni Novgorod, Russia, June 2-5, 1997, 91-92.

The dvi file for this article is available here

------ ------ ------

Extension and Approximation of CR functions on tube manifolds
by A. Boivin and R. Dwilewicz

Abstract: In this paper we prove the following theorem: Any continuous CR function on a tube manifold t(N) = N x iRⁿ in Cⁿ, where N is an embedded submanifold of Rⁿ of class C², can be continuously CR extended to the tube over the convex hull of N. There are no additional assumptions on N, and in this sense, the theorem generalizes all known results of this type, including the classical Bochner tube theorem. Also it implies almost uniform approximation of CR functions on tubes by holomorphic polynomials in Cⁿ.

Trans. Amer. Math. Soc. 350, 1945-1956 (1998)

The dvi file for this article is available here

------ ------ ------

An example in tangential meromorphic approximation
by A. Boivin and A.H. Nersessian

Abstract: It is known that all sets of meromorphic uniform approximation in C which satisfy an additional condition involving the Gleason parts of the algebra R(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, though it is a set of tangential approximation, does not satisfy this extra condition on parts, and thus showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem.
© 1996 Academic Press Inc.

Journal of Approximation Theory 87, 103-111 (1996)

------ ------ ------

Meromorphic approximation in weighted L^p spaces
by A. Boivin, A. Bonilla and J.C. Fariña

Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995)
------ ------ ------

On the L²-completeness of some systems of analytic functions
by A. Boivin and C. Zhu

Abstract: For an entire function f(z), a sequence of complex numbers {µ_n} and an unbounded domain G, the completeness of the system {f(µ_nz)} in L²_a(G) (mean-square approximation) is studied. A similar problem is discussed when f is analytic on the Riemann surface of the logarithm.
© 2001 OPA (Overseas Publishers Association) N.V.

Complex Variables - Theory and Application, 45, 273-295 (2001)

------ ------ ------

Holomorphic and harmonic approximation on Riemann surfaces
by A. Boivin and P.M. Gauthier

Abstract: The most important result on open Riemann surfaces is the analog of Runge's theorem on approximation by complex polynomials, due to Behnke and Stein. In this course, we begin by a discussion of Runge's theorems on polynomial and rational approximation on compact sets. This theory is refined and extended in various ways to Riemann surfaces. We also introduce a corresponding theory of harmonic approximation.
© 2001 Kluwer Academic Publishers

Approximation, Complex Analysis and Potential Theory
Proceedings of a NATO ASI held in Montreal, July 2000
N. Arakelian and P.M. Gauthier, eds.
NATO Science Series II, Vol. 37
Kluwer Academic Publishers 2001, 107--128

------ ------ ------

On radial limit functions for entire solutions of second order elliptic equations in R²
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L of order two with constant complex coefficients in R², we consider entire solutions of the equation Lu = 0 for which U(e^it) := lim u(re^it) exists for all t in [0, 2pi) as a finite limit in C as r goes to infinity. We characterize the possible ``radial limit functions" U. This is an analog of the work of Alice Roth for entire holomorphic functions. The results seem new even for harmonic functions.

Publicacions Matemàtiques, 42, No.2, 509-519 (1998)

The dvi file for this article is available here

------ ------ ------

Approximation by meromorphic and entire solutions
of elliptic equations in Banach spaces of distributions
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant coefficients and a class of functions (jets-distributions) which are defined on a closed, not necessarily compact, subset of Rⁿ and which belong locally to a Banach space V, we consider the approximation in the norm of V of the functions (jets) in this class by global (entire or meromorphic) solutions of the equation Lu=0. We establish Runge, Mergelyan, Roth and Arakelyan - type theorems for a large class of Banach spaces V and operators L, encompassing most of the previously considered generalizations of these theorems and obtaining new ones.

Sbornik Math. 189, No.4, 481--502 (1998) (in English)
Mat. Sbornik 189, No.4, 3--24 (1998) (in Russian)

The dvi file for this article (in English) is available here

------ ------ ------

Approximation by entire functions
by A. Boivin and P.V. Paramonov

Abstract: Examples of the type of (new) results we can deduce from the theorems obtained in the paper "Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions" are given in the case when L is the Cauchy-Riemann or the Laplacian operator and the norm is the C^m norm.

Proceeding of a Conference in memory of A.F. Leont'ev, Nizhni Novgorod, Russia, June 2-5, 1997, 91-92.

The dvi file for this article is available here

------ ------ ------

Extension and Approximation of CR functions on tube manifolds
by A. Boivin and R. Dwilewicz

Abstract: In this paper we prove the following theorem: Any continuous CR function on a tube manifold t(N) = N x iRⁿ in Cⁿ, where N is an embedded submanifold of Rⁿ of class C², can be continuously CR extended to the tube over the convex hull of N. There are no additional assumptions on N, and in this sense, the theorem generalizes all known results of this type, including the classical Bochner tube theorem. Also it implies almost uniform approximation of CR functions on tubes by holomorphic polynomials in Cⁿ.

Trans. Amer. Math. Soc. 350, 1945-1956 (1998)

The dvi file for this article is available here

------ ------ ------

An example in tangential meromorphic approximation
by A. Boivin and A.H. Nersessian

Abstract: It is known that all sets of meromorphic uniform approximation in C which satisfy an additional condition involving the Gleason parts of the algebra R(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, though it is a set of tangential approximation, does not satisfy this extra condition on parts, and thus showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem.
© 1996 Academic Press Inc.

Journal of Approximation Theory 87, 103-111 (1996)

------ ------ ------

Meromorphic approximation in weighted L^p spaces
by A. Boivin, A. Bonilla and J.C. Fariña

Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995): Abstract: For an entire function f(z), a sequence of complex numbers {µ_n} and an unbounded domain G, the completeness of the system {f(µ_nz)} in L²_a(G) (mean-square approximation) is studied. A similar problem is discussed when f is analytic on the Riemann surface of the logarithm.
© 2001 OPA (Overseas Publishers Association) N.V.

Complex Variables - Theory and Application, 45, 273-295 (2001)

------ ------ ------

Holomorphic and harmonic approximation on Riemann surfaces
by A. Boivin and P.M. Gauthier

Abstract: The most important result on open Riemann surfaces is the analog of Runge's theorem on approximation by complex polynomials, due to Behnke and Stein. In this course, we begin by a discussion of Runge's theorems on polynomial and rational approximation on compact sets. This theory is refined and extended in various ways to Riemann surfaces. We also introduce a corresponding theory of harmonic approximation.
© 2001 Kluwer Academic Publishers

Approximation, Complex Analysis and Potential Theory
Proceedings of a NATO ASI held in Montreal, July 2000
N. Arakelian and P.M. Gauthier, eds.
NATO Science Series II, Vol. 37
Kluwer Academic Publishers 2001, 107--128

------ ------ ------

On radial limit functions for entire solutions of second order elliptic equations in R²
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L of order two with constant complex coefficients in R², we consider entire solutions of the equation Lu = 0 for which U(e^it) := lim u(re^it) exists for all t in [0, 2pi) as a finite limit in C as r goes to infinity. We characterize the possible ``radial limit functions" U. This is an analog of the work of Alice Roth for entire holomorphic functions. The results seem new even for harmonic functions.

Publicacions Matemàtiques, 42, No.2, 509-519 (1998)

The dvi file for this article is available here

------ ------ ------

Approximation by meromorphic and entire solutions
of elliptic equations in Banach spaces of distributions
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant coefficients and a class of functions (jets-distributions) which are defined on a closed, not necessarily compact, subset of Rⁿ and which belong locally to a Banach space V, we consider the approximation in the norm of V of the functions (jets) in this class by global (entire or meromorphic) solutions of the equation Lu=0. We establish Runge, Mergelyan, Roth and Arakelyan - type theorems for a large class of Banach spaces V and operators L, encompassing most of the previously considered generalizations of these theorems and obtaining new ones.

Sbornik Math. 189, No.4, 481--502 (1998) (in English)
Mat. Sbornik 189, No.4, 3--24 (1998) (in Russian)

The dvi file for this article (in English) is available here

------ ------ ------

Approximation by entire functions
by A. Boivin and P.V. Paramonov

Abstract: Examples of the type of (new) results we can deduce from the theorems obtained in the paper "Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions" are given in the case when L is the Cauchy-Riemann or the Laplacian operator and the norm is the C^m norm.

Proceeding of a Conference in memory of A.F. Leont'ev, Nizhni Novgorod, Russia, June 2-5, 1997, 91-92.

The dvi file for this article is available here

------ ------ ------

Extension and Approximation of CR functions on tube manifolds
by A. Boivin and R. Dwilewicz

Abstract: In this paper we prove the following theorem: Any continuous CR function on a tube manifold t(N) = N x iRⁿ in Cⁿ, where N is an embedded submanifold of Rⁿ of class C², can be continuously CR extended to the tube over the convex hull of N. There are no additional assumptions on N, and in this sense, the theorem generalizes all known results of this type, including the classical Bochner tube theorem. Also it implies almost uniform approximation of CR functions on tubes by holomorphic polynomials in Cⁿ.

Trans. Amer. Math. Soc. 350, 1945-1956 (1998)

The dvi file for this article is available here

------ ------ ------

An example in tangential meromorphic approximation
by A. Boivin and A.H. Nersessian

Abstract: It is known that all sets of meromorphic uniform approximation in C which satisfy an additional condition involving the Gleason parts of the algebra R(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, though it is a set of tangential approximation, does not satisfy this extra condition on parts, and thus showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem.
© 1996 Academic Press Inc.

Journal of Approximation Theory 87, 103-111 (1996)

------ ------ ------

Meromorphic approximation in weighted L^p spaces
by A. Boivin, A. Bonilla and J.C. Fariña

Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995)
------ ------ ------

Holomorphic and harmonic approximation on Riemann surfaces
by A. Boivin and P.M. Gauthier

Abstract: The most important result on open Riemann surfaces is the analog of Runge's theorem on approximation by complex polynomials, due to Behnke and Stein. In this course, we begin by a discussion of Runge's theorems on polynomial and rational approximation on compact sets. This theory is refined and extended in various ways to Riemann surfaces. We also introduce a corresponding theory of harmonic approximation.
© 2001 Kluwer Academic Publishers

Approximation, Complex Analysis and Potential Theory
Proceedings of a NATO ASI held in Montreal, July 2000
N. Arakelian and P.M. Gauthier, eds.
NATO Science Series II, Vol. 37
Kluwer Academic Publishers 2001, 107--128

------ ------ ------

On radial limit functions for entire solutions of second order elliptic equations in R²
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L of order two with constant complex coefficients in R², we consider entire solutions of the equation Lu = 0 for which U(e^it) := lim u(re^it) exists for all t in [0, 2pi) as a finite limit in C as r goes to infinity. We characterize the possible ``radial limit functions" U. This is an analog of the work of Alice Roth for entire holomorphic functions. The results seem new even for harmonic functions.

Publicacions Matemàtiques, 42, No.2, 509-519 (1998)

The dvi file for this article is available here

------ ------ ------

Approximation by meromorphic and entire solutions
of elliptic equations in Banach spaces of distributions
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant coefficients and a class of functions (jets-distributions) which are defined on a closed, not necessarily compact, subset of Rⁿ and which belong locally to a Banach space V, we consider the approximation in the norm of V of the functions (jets) in this class by global (entire or meromorphic) solutions of the equation Lu=0. We establish Runge, Mergelyan, Roth and Arakelyan - type theorems for a large class of Banach spaces V and operators L, encompassing most of the previously considered generalizations of these theorems and obtaining new ones.

Sbornik Math. 189, No.4, 481--502 (1998) (in English)
Mat. Sbornik 189, No.4, 3--24 (1998) (in Russian)

The dvi file for this article (in English) is available here

------ ------ ------

Approximation by entire functions
by A. Boivin and P.V. Paramonov

Abstract: Examples of the type of (new) results we can deduce from the theorems obtained in the paper "Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions" are given in the case when L is the Cauchy-Riemann or the Laplacian operator and the norm is the C^m norm.

Proceeding of a Conference in memory of A.F. Leont'ev, Nizhni Novgorod, Russia, June 2-5, 1997, 91-92.

The dvi file for this article is available here

------ ------ ------

Extension and Approximation of CR functions on tube manifolds
by A. Boivin and R. Dwilewicz

Abstract: In this paper we prove the following theorem: Any continuous CR function on a tube manifold t(N) = N x iRⁿ in Cⁿ, where N is an embedded submanifold of Rⁿ of class C², can be continuously CR extended to the tube over the convex hull of N. There are no additional assumptions on N, and in this sense, the theorem generalizes all known results of this type, including the classical Bochner tube theorem. Also it implies almost uniform approximation of CR functions on tubes by holomorphic polynomials in Cⁿ.

Trans. Amer. Math. Soc. 350, 1945-1956 (1998)

The dvi file for this article is available here

------ ------ ------

An example in tangential meromorphic approximation
by A. Boivin and A.H. Nersessian

Abstract: It is known that all sets of meromorphic uniform approximation in C which satisfy an additional condition involving the Gleason parts of the algebra R(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, though it is a set of tangential approximation, does not satisfy this extra condition on parts, and thus showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem.
© 1996 Academic Press Inc.

Journal of Approximation Theory 87, 103-111 (1996)

------ ------ ------

Meromorphic approximation in weighted L^p spaces
by A. Boivin, A. Bonilla and J.C. Fariña

Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995): Abstract: The most important result on open Riemann surfaces is the analog of Runge's theorem on approximation by complex polynomials, due to Behnke and Stein. In this course, we begin by a discussion of Runge's theorems on polynomial and rational approximation on compact sets. This theory is refined and extended in various ways to Riemann surfaces. We also introduce a corresponding theory of harmonic approximation.
© 2001 Kluwer Academic Publishers

Approximation, Complex Analysis and Potential Theory
Proceedings of a NATO ASI held in Montreal, July 2000
N. Arakelian and P.M. Gauthier, eds.
NATO Science Series II, Vol. 37
Kluwer Academic Publishers 2001, 107--128

------ ------ ------

On radial limit functions for entire solutions of second order elliptic equations in R²
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L of order two with constant complex coefficients in R², we consider entire solutions of the equation Lu = 0 for which U(e^it) := lim u(re^it) exists for all t in [0, 2pi) as a finite limit in C as r goes to infinity. We characterize the possible ``radial limit functions" U. This is an analog of the work of Alice Roth for entire holomorphic functions. The results seem new even for harmonic functions.

Publicacions Matemàtiques, 42, No.2, 509-519 (1998)

The dvi file for this article is available here

------ ------ ------

Approximation by meromorphic and entire solutions
of elliptic equations in Banach spaces of distributions
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant coefficients and a class of functions (jets-distributions) which are defined on a closed, not necessarily compact, subset of Rⁿ and which belong locally to a Banach space V, we consider the approximation in the norm of V of the functions (jets) in this class by global (entire or meromorphic) solutions of the equation Lu=0. We establish Runge, Mergelyan, Roth and Arakelyan - type theorems for a large class of Banach spaces V and operators L, encompassing most of the previously considered generalizations of these theorems and obtaining new ones.

Sbornik Math. 189, No.4, 481--502 (1998) (in English)
Mat. Sbornik 189, No.4, 3--24 (1998) (in Russian)

The dvi file for this article (in English) is available here

------ ------ ------

Approximation by entire functions
by A. Boivin and P.V. Paramonov

Abstract: Examples of the type of (new) results we can deduce from the theorems obtained in the paper "Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions" are given in the case when L is the Cauchy-Riemann or the Laplacian operator and the norm is the C^m norm.

Proceeding of a Conference in memory of A.F. Leont'ev, Nizhni Novgorod, Russia, June 2-5, 1997, 91-92.

The dvi file for this article is available here

------ ------ ------

Extension and Approximation of CR functions on tube manifolds
by A. Boivin and R. Dwilewicz

Abstract: In this paper we prove the following theorem: Any continuous CR function on a tube manifold t(N) = N x iRⁿ in Cⁿ, where N is an embedded submanifold of Rⁿ of class C², can be continuously CR extended to the tube over the convex hull of N. There are no additional assumptions on N, and in this sense, the theorem generalizes all known results of this type, including the classical Bochner tube theorem. Also it implies almost uniform approximation of CR functions on tubes by holomorphic polynomials in Cⁿ.

Trans. Amer. Math. Soc. 350, 1945-1956 (1998)

The dvi file for this article is available here

------ ------ ------

An example in tangential meromorphic approximation
by A. Boivin and A.H. Nersessian

Abstract: It is known that all sets of meromorphic uniform approximation in C which satisfy an additional condition involving the Gleason parts of the algebra R(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, though it is a set of tangential approximation, does not satisfy this extra condition on parts, and thus showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem.
© 1996 Academic Press Inc.

Journal of Approximation Theory 87, 103-111 (1996)

------ ------ ------

Meromorphic approximation in weighted L^p spaces
by A. Boivin, A. Bonilla and J.C. Fariña

Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995)
------ ------ ------

On radial limit functions for entire solutions of second order elliptic equations in R²
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L of order two with constant complex coefficients in R², we consider entire solutions of the equation Lu = 0 for which U(e^it) := lim u(re^it) exists for all t in [0, 2pi) as a finite limit in C as r goes to infinity. We characterize the possible ``radial limit functions" U. This is an analog of the work of Alice Roth for entire holomorphic functions. The results seem new even for harmonic functions.

Publicacions Matemàtiques, 42, No.2, 509-519 (1998)

The dvi file for this article is available here

------ ------ ------

Approximation by meromorphic and entire solutions
of elliptic equations in Banach spaces of distributions
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant coefficients and a class of functions (jets-distributions) which are defined on a closed, not necessarily compact, subset of Rⁿ and which belong locally to a Banach space V, we consider the approximation in the norm of V of the functions (jets) in this class by global (entire or meromorphic) solutions of the equation Lu=0. We establish Runge, Mergelyan, Roth and Arakelyan - type theorems for a large class of Banach spaces V and operators L, encompassing most of the previously considered generalizations of these theorems and obtaining new ones.

Sbornik Math. 189, No.4, 481--502 (1998) (in English)
Mat. Sbornik 189, No.4, 3--24 (1998) (in Russian)

The dvi file for this article (in English) is available here

------ ------ ------

Approximation by entire functions
by A. Boivin and P.V. Paramonov

Abstract: Examples of the type of (new) results we can deduce from the theorems obtained in the paper "Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions" are given in the case when L is the Cauchy-Riemann or the Laplacian operator and the norm is the C^m norm.

Proceeding of a Conference in memory of A.F. Leont'ev, Nizhni Novgorod, Russia, June 2-5, 1997, 91-92.

The dvi file for this article is available here

------ ------ ------

Extension and Approximation of CR functions on tube manifolds
by A. Boivin and R. Dwilewicz

Abstract: In this paper we prove the following theorem: Any continuous CR function on a tube manifold t(N) = N x iRⁿ in Cⁿ, where N is an embedded submanifold of Rⁿ of class C², can be continuously CR extended to the tube over the convex hull of N. There are no additional assumptions on N, and in this sense, the theorem generalizes all known results of this type, including the classical Bochner tube theorem. Also it implies almost uniform approximation of CR functions on tubes by holomorphic polynomials in Cⁿ.

Trans. Amer. Math. Soc. 350, 1945-1956 (1998)

The dvi file for this article is available here

------ ------ ------

An example in tangential meromorphic approximation
by A. Boivin and A.H. Nersessian

Abstract: It is known that all sets of meromorphic uniform approximation in C which satisfy an additional condition involving the Gleason parts of the algebra R(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, though it is a set of tangential approximation, does not satisfy this extra condition on parts, and thus showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem.
© 1996 Academic Press Inc.

Journal of Approximation Theory 87, 103-111 (1996)

------ ------ ------

Meromorphic approximation in weighted L^p spaces
by A. Boivin, A. Bonilla and J.C. Fariña

Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995): Abstract: Given a homogeneous elliptic partial differential operator L of order two with constant complex coefficients in R², we consider entire solutions of the equation Lu = 0 for which U(e^it) := lim u(re^it) exists for all t in [0, 2pi) as a finite limit in C as r goes to infinity. We characterize the possible ``radial limit functions" U. This is an analog of the work of Alice Roth for entire holomorphic functions. The results seem new even for harmonic functions.

Publicacions Matemàtiques, 42, No.2, 509-519 (1998)

The dvi file for this article is available here

------ ------ ------

Approximation by meromorphic and entire solutions
of elliptic equations in Banach spaces of distributions
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant coefficients and a class of functions (jets-distributions) which are defined on a closed, not necessarily compact, subset of Rⁿ and which belong locally to a Banach space V, we consider the approximation in the norm of V of the functions (jets) in this class by global (entire or meromorphic) solutions of the equation Lu=0. We establish Runge, Mergelyan, Roth and Arakelyan - type theorems for a large class of Banach spaces V and operators L, encompassing most of the previously considered generalizations of these theorems and obtaining new ones.

Sbornik Math. 189, No.4, 481--502 (1998) (in English)
Mat. Sbornik 189, No.4, 3--24 (1998) (in Russian)

The dvi file for this article (in English) is available here

------ ------ ------

Approximation by entire functions
by A. Boivin and P.V. Paramonov

Abstract: Examples of the type of (new) results we can deduce from the theorems obtained in the paper "Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions" are given in the case when L is the Cauchy-Riemann or the Laplacian operator and the norm is the C^m norm.

Proceeding of a Conference in memory of A.F. Leont'ev, Nizhni Novgorod, Russia, June 2-5, 1997, 91-92.

The dvi file for this article is available here

------ ------ ------

Extension and Approximation of CR functions on tube manifolds
by A. Boivin and R. Dwilewicz

Abstract: In this paper we prove the following theorem: Any continuous CR function on a tube manifold t(N) = N x iRⁿ in Cⁿ, where N is an embedded submanifold of Rⁿ of class C², can be continuously CR extended to the tube over the convex hull of N. There are no additional assumptions on N, and in this sense, the theorem generalizes all known results of this type, including the classical Bochner tube theorem. Also it implies almost uniform approximation of CR functions on tubes by holomorphic polynomials in Cⁿ.

Trans. Amer. Math. Soc. 350, 1945-1956 (1998)

The dvi file for this article is available here

------ ------ ------

An example in tangential meromorphic approximation
by A. Boivin and A.H. Nersessian

Abstract: It is known that all sets of meromorphic uniform approximation in C which satisfy an additional condition involving the Gleason parts of the algebra R(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, though it is a set of tangential approximation, does not satisfy this extra condition on parts, and thus showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem.
© 1996 Academic Press Inc.

Journal of Approximation Theory 87, 103-111 (1996)

------ ------ ------

Meromorphic approximation in weighted L^p spaces
by A. Boivin, A. Bonilla and J.C. Fariña

Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995)
------ ------ ------

Approximation by meromorphic and entire solutions
of elliptic equations in Banach spaces of distributions
by A. Boivin and P.V. Paramonov

Abstract: Given a homogeneous elliptic partial differential operator L with constant coefficients and a class of functions (jets-distributions) which are defined on a closed, not necessarily compact, subset of Rⁿ and which belong locally to a Banach space V, we consider the approximation in the norm of V of the functions (jets) in this class by global (entire or meromorphic) solutions of the equation Lu=0. We establish Runge, Mergelyan, Roth and Arakelyan - type theorems for a large class of Banach spaces V and operators L, encompassing most of the previously considered generalizations of these theorems and obtaining new ones.

Sbornik Math. 189, No.4, 481--502 (1998) (in English)
Mat. Sbornik 189, No.4, 3--24 (1998) (in Russian)

The dvi file for this article (in English) is available here

------ ------ ------

Approximation by entire functions
by A. Boivin and P.V. Paramonov

Abstract: Examples of the type of (new) results we can deduce from the theorems obtained in the paper "Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions" are given in the case when L is the Cauchy-Riemann or the Laplacian operator and the norm is the C^m norm.

Proceeding of a Conference in memory of A.F. Leont'ev, Nizhni Novgorod, Russia, June 2-5, 1997, 91-92.

The dvi file for this article is available here

------ ------ ------

Extension and Approximation of CR functions on tube manifolds
by A. Boivin and R. Dwilewicz

Abstract: In this paper we prove the following theorem: Any continuous CR function on a tube manifold t(N) = N x iRⁿ in Cⁿ, where N is an embedded submanifold of Rⁿ of class C², can be continuously CR extended to the tube over the convex hull of N. There are no additional assumptions on N, and in this sense, the theorem generalizes all known results of this type, including the classical Bochner tube theorem. Also it implies almost uniform approximation of CR functions on tubes by holomorphic polynomials in Cⁿ.

Trans. Amer. Math. Soc. 350, 1945-1956 (1998)

The dvi file for this article is available here

------ ------ ------

An example in tangential meromorphic approximation
by A. Boivin and A.H. Nersessian

Abstract: It is known that all sets of meromorphic uniform approximation in C which satisfy an additional condition involving the Gleason parts of the algebra R(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, though it is a set of tangential approximation, does not satisfy this extra condition on parts, and thus showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem.
© 1996 Academic Press Inc.

Journal of Approximation Theory 87, 103-111 (1996)

------ ------ ------

Meromorphic approximation in weighted L^p spaces
by A. Boivin, A. Bonilla and J.C. Fariña

Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995): Abstract: Given a homogeneous elliptic partial differential operator L with constant coefficients and a class of functions (jets-distributions) which are defined on a closed, not necessarily compact, subset of Rⁿ and which belong locally to a Banach space V, we consider the approximation in the norm of V of the functions (jets) in this class by global (entire or meromorphic) solutions of the equation Lu=0. We establish Runge, Mergelyan, Roth and Arakelyan - type theorems for a large class of Banach spaces V and operators L, encompassing most of the previously considered generalizations of these theorems and obtaining new ones.

Sbornik Math. 189, No.4, 481--502 (1998) (in English)
Mat. Sbornik 189, No.4, 3--24 (1998) (in Russian)

The dvi file for this article (in English) is available here

------ ------ ------

Approximation by entire functions
by A. Boivin and P.V. Paramonov

Abstract: Examples of the type of (new) results we can deduce from the theorems obtained in the paper "Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions" are given in the case when L is the Cauchy-Riemann or the Laplacian operator and the norm is the C^m norm.

Proceeding of a Conference in memory of A.F. Leont'ev, Nizhni Novgorod, Russia, June 2-5, 1997, 91-92.

The dvi file for this article is available here

------ ------ ------

Extension and Approximation of CR functions on tube manifolds
by A. Boivin and R. Dwilewicz

Abstract: In this paper we prove the following theorem: Any continuous CR function on a tube manifold t(N) = N x iRⁿ in Cⁿ, where N is an embedded submanifold of Rⁿ of class C², can be continuously CR extended to the tube over the convex hull of N. There are no additional assumptions on N, and in this sense, the theorem generalizes all known results of this type, including the classical Bochner tube theorem. Also it implies almost uniform approximation of CR functions on tubes by holomorphic polynomials in Cⁿ.

Trans. Amer. Math. Soc. 350, 1945-1956 (1998)

The dvi file for this article is available here

------ ------ ------

An example in tangential meromorphic approximation
by A. Boivin and A.H. Nersessian

Abstract: It is known that all sets of meromorphic uniform approximation in C which satisfy an additional condition involving the Gleason parts of the algebra R(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, though it is a set of tangential approximation, does not satisfy this extra condition on parts, and thus showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem.
© 1996 Academic Press Inc.

Journal of Approximation Theory 87, 103-111 (1996)

------ ------ ------

Meromorphic approximation in weighted L^p spaces
by A. Boivin, A. Bonilla and J.C. Fariña

Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995)
------ ------ ------

Approximation by entire functions
by A. Boivin and P.V. Paramonov

Abstract: Examples of the type of (new) results we can deduce from the theorems obtained in the paper "Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions" are given in the case when L is the Cauchy-Riemann or the Laplacian operator and the norm is the C^m norm.

Proceeding of a Conference in memory of A.F. Leont'ev, Nizhni Novgorod, Russia, June 2-5, 1997, 91-92.

The dvi file for this article is available here

------ ------ ------

Extension and Approximation of CR functions on tube manifolds
by A. Boivin and R. Dwilewicz

Abstract: In this paper we prove the following theorem: Any continuous CR function on a tube manifold t(N) = N x iRⁿ in Cⁿ, where N is an embedded submanifold of Rⁿ of class C², can be continuously CR extended to the tube over the convex hull of N. There are no additional assumptions on N, and in this sense, the theorem generalizes all known results of this type, including the classical Bochner tube theorem. Also it implies almost uniform approximation of CR functions on tubes by holomorphic polynomials in Cⁿ.

Trans. Amer. Math. Soc. 350, 1945-1956 (1998)

The dvi file for this article is available here

------ ------ ------

An example in tangential meromorphic approximation
by A. Boivin and A.H. Nersessian

Abstract: It is known that all sets of meromorphic uniform approximation in C which satisfy an additional condition involving the Gleason parts of the algebra R(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, though it is a set of tangential approximation, does not satisfy this extra condition on parts, and thus showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem.
© 1996 Academic Press Inc.

Journal of Approximation Theory 87, 103-111 (1996)

------ ------ ------

Meromorphic approximation in weighted L^p spaces
by A. Boivin, A. Bonilla and J.C. Fariña

Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995): Abstract: Examples of the type of (new) results we can deduce from the theorems obtained in the paper "Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions" are given in the case when L is the Cauchy-Riemann or the Laplacian operator and the norm is the C^m norm.

Proceeding of a Conference in memory of A.F. Leont'ev, Nizhni Novgorod, Russia, June 2-5, 1997, 91-92.

The dvi file for this article is available here

------ ------ ------

Extension and Approximation of CR functions on tube manifolds
by A. Boivin and R. Dwilewicz

Abstract: In this paper we prove the following theorem: Any continuous CR function on a tube manifold t(N) = N x iRⁿ in Cⁿ, where N is an embedded submanifold of Rⁿ of class C², can be continuously CR extended to the tube over the convex hull of N. There are no additional assumptions on N, and in this sense, the theorem generalizes all known results of this type, including the classical Bochner tube theorem. Also it implies almost uniform approximation of CR functions on tubes by holomorphic polynomials in Cⁿ.

Trans. Amer. Math. Soc. 350, 1945-1956 (1998)

The dvi file for this article is available here

------ ------ ------

An example in tangential meromorphic approximation
by A. Boivin and A.H. Nersessian

Abstract: It is known that all sets of meromorphic uniform approximation in C which satisfy an additional condition involving the Gleason parts of the algebra R(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, though it is a set of tangential approximation, does not satisfy this extra condition on parts, and thus showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem.
© 1996 Academic Press Inc.

Journal of Approximation Theory 87, 103-111 (1996)

------ ------ ------

Meromorphic approximation in weighted L^p spaces
by A. Boivin, A. Bonilla and J.C. Fariña

Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995)
------ ------ ------

Extension and Approximation of CR functions on tube manifolds
by A. Boivin and R. Dwilewicz

Abstract: In this paper we prove the following theorem: Any continuous CR function on a tube manifold t(N) = N x iRⁿ in Cⁿ, where N is an embedded submanifold of Rⁿ of class C², can be continuously CR extended to the tube over the convex hull of N. There are no additional assumptions on N, and in this sense, the theorem generalizes all known results of this type, including the classical Bochner tube theorem. Also it implies almost uniform approximation of CR functions on tubes by holomorphic polynomials in Cⁿ.

Trans. Amer. Math. Soc. 350, 1945-1956 (1998)

The dvi file for this article is available here

------ ------ ------

An example in tangential meromorphic approximation
by A. Boivin and A.H. Nersessian

Abstract: It is known that all sets of meromorphic uniform approximation in C which satisfy an additional condition involving the Gleason parts of the algebra R(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, though it is a set of tangential approximation, does not satisfy this extra condition on parts, and thus showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem.
© 1996 Academic Press Inc.

Journal of Approximation Theory 87, 103-111 (1996)

------ ------ ------

Meromorphic approximation in weighted L^p spaces
by A. Boivin, A. Bonilla and J.C. Fariña

Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995): Abstract: In this paper we prove the following theorem: Any continuous CR function on a tube manifold t(N) = N x iRⁿ in Cⁿ, where N is an embedded submanifold of Rⁿ of class C², can be continuously CR extended to the tube over the convex hull of N. There are no additional assumptions on N, and in this sense, the theorem generalizes all known results of this type, including the classical Bochner tube theorem. Also it implies almost uniform approximation of CR functions on tubes by holomorphic polynomials in Cⁿ.

Trans. Amer. Math. Soc. 350, 1945-1956 (1998)

The dvi file for this article is available here

------ ------ ------

An example in tangential meromorphic approximation
by A. Boivin and A.H. Nersessian

Abstract: It is known that all sets of meromorphic uniform approximation in C which satisfy an additional condition involving the Gleason parts of the algebra R(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, though it is a set of tangential approximation, does not satisfy this extra condition on parts, and thus showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem.
© 1996 Academic Press Inc.

Journal of Approximation Theory 87, 103-111 (1996)

------ ------ ------

Meromorphic approximation in weighted L^p spaces
by A. Boivin, A. Bonilla and J.C. Fariña

Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995)
------ ------ ------

An example in tangential meromorphic approximation
by A. Boivin and A.H. Nersessian

Abstract: It is known that all sets of meromorphic uniform approximation in C which satisfy an additional condition involving the Gleason parts of the algebra R(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, though it is a set of tangential approximation, does not satisfy this extra condition on parts, and thus showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem.
© 1996 Academic Press Inc.

Journal of Approximation Theory 87, 103-111 (1996)

------ ------ ------

Meromorphic approximation in weighted L^p spaces
by A. Boivin, A. Bonilla and J.C. Fariña

Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995): Abstract: It is known that all sets of meromorphic uniform approximation in C which satisfy an additional condition involving the Gleason parts of the algebra R(K) are then also sets of tangential approximation by meromorphic functions. In this paper, we construct a set which, though it is a set of tangential approximation, does not satisfy this extra condition on parts, and thus showing that the condition fails to be necessary. Finding a complete characterization of sets of meromorphic tangential approximation is still an open problem.
© 1996 Academic Press Inc.

Journal of Approximation Theory 87, 103-111 (1996)

------ ------ ------

Meromorphic approximation in weighted L^p spaces
by A. Boivin, A. Bonilla and J.C. Fariña

Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995)
------ ------ ------

Meromorphic approximation in weighted L^p spaces
by A. Boivin, A. Bonilla and J.C. Fariña

Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995): Abstract: We study L^p approximation on (unbounded) closed subsets of C. Using an appropriate version of the Fusion Lemma of Alice Roth, we prove a localisation theorem and obtain Runge type and Vitushkin type (i.e. with analytic capacities) theorems on L^p(µ) spaces whenever dµ = wdxdy and w is in the Muckenhoupt class A_p.
© 1995 Royal Irish Academy

Proceedings of the Royal Irish Academy 95A, 47-64 (1995)

Last update: June 2003

Preprints and abstracts: André Boivin

List of Publications:

Abstract of recents papers and preprints:

Uniform approximation on closed subsets of C by polyanalytic functions

Completeness of systems of exponentials and Lambert W functions

Bases of exponentials in weighted L^p spaces

Uniform approximation by meromorphic functions on Riemann surfaces

The growth of an entire function and its Dirichlet coefficients and exponents.

On the completeness of the system { z ^t_n} in L².

Approximation on closed sets by analytic or meromorphic solutions
of elliptic equations and applications

On the L²-completeness of some systems of analytic functions

Holomorphic and harmonic approximation on Riemann surfaces

On radial limit functions for entire solutions of second order elliptic equations in R²

Approximation by meromorphic and entire solutions
of elliptic equations in Banach spaces of distributions

Approximation by entire functions

Extension and Approximation of CR functions on tube manifolds

An example in tangential meromorphic approximation

Meromorphic approximation in weighted L^p spaces

Preprints and abstracts: André Boivin

List of Publications:

Abstract of recents papers and preprints:

Uniform approximation on closed subsets of C by polyanalytic functions

Completeness of systems of exponentials and Lambert W functions

Bases of exponentials in weighted Lp spaces

Uniform approximation by meromorphic functions on Riemann surfaces

The growth of an entire function and its Dirichlet coefficients and exponents.

On the completeness of the system { z tn} in L2.

Approximation on closed sets by analytic or meromorphic solutions of elliptic equations and applications

On the L2-completeness of some systems of analytic functions

Holomorphic and harmonic approximation on Riemann surfaces

On radial limit functions for entire solutions of second order elliptic equations in R2

Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions

Approximation by entire functions

Extension and Approximation of CR functions on tube manifolds

An example in tangential meromorphic approximation

Meromorphic approximation in weighted Lp spaces

Bases of exponentials in weighted L^p spaces

On the completeness of the system { z ^t_n} in L².

Approximation on closed sets by analytic or meromorphic solutions
of elliptic equations and applications

On the L²-completeness of some systems of analytic functions

On radial limit functions for entire solutions of second order elliptic equations in R²

Approximation by meromorphic and entire solutions
of elliptic equations in Banach spaces of distributions

Meromorphic approximation in weighted L^p spaces