Abstract:
This paper is a survey of results obtained over the past 20–30 years in the qualitative theory of approximation of functions by holomorphic, harmonic, and polyanalytic functions (and, in particular, by corresponding polynomials) in the norms of Whitney-type spaces Cm on compact subsets of Euclidean spaces.
Bibliography: 120 titles.
Keywords:Cm-approximation by holomorphic, harmonic, and polyanalytic functions; Cm-analytic and Cm-harmonic capacity; s-dimensional Hausdorff content; Vitushkin localization operator; Nevanlinna domains; Dirichlet problem.
Citation:
M. Ya. Mazalov, P. V. Paramonov, K. Yu. Fedorovskiy, “Conditions for Cm-approximability of functions by solutions of elliptic equations”, Russian Math. Surveys, 67:6 (2012), 1023–1068
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\paper Conditions for $C^m$-approximability of functions by solutions of elliptic equations
\jour Russian Math. Surveys
\yr 2012
\vol 67
\issue 6
\pages 1023--1068
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Linking options:
https://www.mathnet.ru/eng/rm9498
https://doi.org/10.1070/RM2012v067n06ABEH004817
https://www.mathnet.ru/eng/rm/v67/i6/p53
This publication is cited in the following 34 articles:
Sorin G. Gal, Irene Sabadini, “Density of Complex and Quaternionic Polyanalytic Polynomials in Polyanalytic Fock Spaces”, Complex Anal. Oper. Theory, 18:1 (2024)
Astamur Bagapsh, Konstantin Fedorovskiy, Maksim Mazalov, “On Dirichlet problem and uniform approximation by solutions of second-order elliptic systems in R2”, Journal of Mathematical Analysis and Applications, 531:1 (2024), 127896
M. Ya. Mazalov, P. V. Paramonov, K. Yu. Fedorovskiy, “Criteria for Cm-approximability of functions by solutions of homogeneous second-order elliptic equations on compact subsets of RN and related capacities”, Russian Math. Surveys, 79:5 (2024), 847–917
P. V. Paramonov, K. Yu. Fedorovskiy, “Explicit form of fundamental solutions to certain elliptic equations and associated B- and C-capacities”, Sb. Math., 214:4 (2023), 550–566
K. Fedorovskiy, “Uniform Approximation by Polynomial Solutions of Elliptic Systems on Boundaries of Carathéodory Domains in R2”, Lobachevskii J Math, 44:4 (2023), 1299
Gal S.G., Sabadini I., “Approximation By Convolution Polyanalytic Operators in the Complex and Quaternionic Compact Unit Balls”, Comput. Methods Funct. Theory, 2022
M. Ya. Mazalov, “Uniform approximation of functions
by solutions of second order homogeneous strongly elliptic equations on compact sets in R2”, Izv. Math., 85:3 (2021), 421–456
P. V. Paramonov, “Criteria for C1-approximability of functions on compact sets in RN, N⩾, by solutions of second-order homogeneous elliptic equations”, Izv. Math., 85:3 (2021), 483–505
P. V. Paramonov, “Uniform approximation of functions by solutions of strongly elliptic equations of second order on compact subsets of \mathbb R^2”, Sb. Math., 212:12 (2021), 1730–1745
Zoubeir H., Kabbaj S., “On the Representation and the Uniform Polynomial Approximation of Polyanalytic Functions of Gevrey Type on the Unit Disk”, Iran. J. Math. Sci. Inform., 16:2 (2021), 89–115
M. Ya. Mazalov, “Approximation by polyanalytic functions in Hölder spaces”, St. Petersburg Math. J., 33:5 (2022), 829–848
P. V. Paramonov, K. Yu. Fedorovskiy, “On C^m-reflection of harmonic functions and C^m-approximation by harmonic polynomials”, Sb. Math., 211:8 (2020), 1159–1170
M. Ya. Mazalov, “A criterion for uniform approximability of individual functions by solutions of second-order homogeneous elliptic equations with constant complex coefficients”, Sb. Math., 211:9 (2020), 1267–1309
Belov Yu. Borichev A. Fedorovskiy K., “Nevanlinna Domains With Large Boundaries”, J. Funct. Anal., 277:8 (2019), 2617–2643
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
Yu. S. Belov, K. Yu. Fedorovskiy, “Model spaces containing univalent functions”, Russian Math. Surveys, 73:1 (2018), 172–174
Fedorovskiy K.Yu., “Two Problems on Approximation By Solutions of Elliptic Systems on Compact Sets in the Plane”, Complex Var. Elliptic Equ., 63:7-8, SI (2018), 961–975
M. Ya. Mazalov, “On Bianalytic Capacities”, Math. Notes, 103:4 (2018), 672–677
P. V. Paramonov, “Criteria for the individual C^m-approximability of functions on compact subsets of \mathbb R^N by solutions of second-order homogeneous elliptic equations”, Sb. Math., 209:6 (2018), 857–870
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