Abstract
In this paper, we study necessary and sufficient conditions for functions to be approximated uniformly on plane compact sets by polynomial solutions to second-order homogeneous elliptic equations with constant coefficients. Sufficient conditions for approximability are of reductive character, i.e., the possibility of approximating on some (simpler) parts of the compact set implies approximability on the entire compact set.
Similar content being viewed by others
REFERENCES
A. B. Zaitsev, “On uniform approximability of functions by polynomials of special classes on compact sets in ℝ2,” Mat. Zametki [Math. Notes], 71 (2002), no. 1, 75–87.
P. V. Paramonov and K. Yu. Fedorovskii, “On uniform and C 1-approximability of functions on compact sets in ℝ2 by solutions of second-order elliptic equations,” Mat. Sb. [Russian Acad. Sci. Sb. Math.], 190 (1999), no. 2, 123–144.
M. Reed and B. Simon, Methods of Modern Mathematical Physics. Functional Analysis, Academic Press, New York, 1972.
Th. Gamelin, Uniform Algebras, Prentice-Hall, Englewood Cliffs (USA), 1969.
E. Bishop, “Boundary measures of analytic differentials,” in: Some Problems of Approximation Theory [in Russian], Inostr. Lit., Moscow, 1963, pp. 87–100.
N. S. Landkof, Fundamentals of Modern Potential Theory [in Russian], Nauka, Moscow, 1966.
K. Yu. Fedorovskii, “On uniform approximations of functions by n-analytic polynomials on rectifiable contours in ℂ,” Mat. Zametki [Math. Notes], 59 (1996), no. 4, 604–610.
J. J. Carmona, K. Yu. Fedorovskii, and P. V. Paramonov, On Uniform Approximation by Polyanalytic Polynomials and the Dirichlet Problem for Bianalytic Functions, Preprint no. 415, Centre de Recerca Matematica, Barcelona. Spain, 1999.
P. V. Paramonov and J. Verdera, “Approximation by solutions of elliptic equations on closed subsets of euclidean space,” Math. Scand., 74 (1994), no. 2, 249–259.
G. M. Goluzin, Geometric Theory of Functions of a Complex Variable [in Russian], Nauka, Moscow, 1966.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Zaitsev, A.B. Uniform Approximation of Functions by Polynomial Solutions to Second-Order Elliptic Equations on Compact Sets in ℝ2 . Mathematical Notes 74, 38–48 (2003). https://doi.org/10.1023/A:1025010914890
Issue Date:
DOI: https://doi.org/10.1023/A:1025010914890