This article is cited in 39 papers

Finding Solution Subspaces of the Laplace and Heat Equations Isometric to Spaces of Real Functions, and Some of Their Applications

D. N. Bushev, Yu. I. Kharkevich

Lesya Ukrainka East European National University

Abstract: We single out subspaces of harmonic functions in the upper half-plane coinciding with spaces of convolutions with the Abel–Poisson kernel and subspaces of solutions of the heat equation coinciding with spaces of convolutions with the Gauss–Weierstrass kernel that are isometric to the corresponding spaces of real functions defined on the set of real numbers. It is shown that, due to isometry, the main approximation characteristics of functions and function classes in these subspaces are equal to the corresponding approximation characteristics of functions and function classes of one variable.

Keywords: Laplace equation, Abel–Poisson delta kernel, Gauss–Weierstrass delta kernel, heat equation, space of convolutions, Lebesgue point, Hölder's inequality.

UDC: 517.5

Received: 02.03.2017
Revised: 05.08.2017

DOI: 10.4213/mzm11573

 English version: Mathematical Notes, 2018, 103:6, 869–880

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