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 Mat. Sb., 1998, Volume 189, Number 4, Pages 3–24 (Mi msb303)

Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions

A. Boivina, P. V. Paramonovb

a University of Western Ontario, Department of Mathematics
b M. V. Lomonosov Moscow State University

Abstract: For a homogeneous elliptic partial differential operator $L$ with constant coefficients and a class of functions (jet-distributions) defined on a closed, not necessarily compact, subset of $\mathbb R^n$ and belonging locally to a Banach space $V$, the approximation in the norm of $V$ of functions in this class by entire and meromorphic solutions of the equation $Lu=0$ is considered. Theorems of Runge, Mergelyan, Roth, and Arakelyan type are established for a wide class of Banach spaces $V$ and operators $L$ they encompass most of the previously considered generalizations of these theorems but also include new results.

DOI: https://doi.org/10.4213/sm303

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English version:
Sbornik: Mathematics, 1998, 189:4, 481–502

Bibliographic databases:

UDC: 517.538.5+517.956.2
MSC: 30E10, 35Jxx

Citation: A. Boivin, P. V. Paramonov, “Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions”, Mat. Sb., 189:4 (1998), 3–24; Sb. Math., 189:4 (1998), 481–502

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Bonilla, A, “Lip alpha harmonic approximation on closed sets”, Proceedings of the American Mathematical Society, 129:9 (2001), 2741
2. J. J. Carmona, P. V. Paramonov, K. Yu. Fedorovskiy, “On uniform approximation by polyanalytic polynomials and the Dirichlet problem for bianalytic functions”, Sb. Math., 193:10 (2002), 1469–1492
3. Boivin, A, “Approximation on closed sets by analytic or meromorphic solutions of elliptic equations and applications”, Canadian Journal of Mathematics-Journal Canadien de Mathematiques, 54:5 (2002), 945
4. A. M. Voroncov, “Joint Approximations of Distributions in Banach Spaces”, Math. Notes, 73:2 (2003), 168–182
5. A. M. Voroncov, “Estimates of $C^m$-Capacity of Compact Sets in $\mathbb{R}^N$”, Math. Notes, 75:6 (2004), 751–764
6. K. Yu. Fedorovskiy, “On Some Properties and Examples of Nevanlinna Domains”, Proc. Steklov Inst. Math., 253 (2006), 186–194
7. Bernal-Gonzalez, L, “Maximal cluster sets of L-analytic functions along arbitrary curves”, Constructive Approximation, 25:2 (2007), 211
8. M. Ya. Mazalov, P. V. Paramonov, K. Yu. Fedorovskiy, “Conditions for $C^m$-approximability of functions by solutions of elliptic equations”, Russian Math. Surveys, 67:6 (2012), 1023–1068
9. Paramonov P.V., Tolsa X., “On C-1-Approximability of Functions By Solutions of Second Order Elliptic Equations on Plane Compact Sets and C-Analytic Capacity”, Anal. Math. Phys., 9:3 (2019), 1133–1161
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