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.
Similar content being viewed by others
References
D. M. Bushev, “Isometry of the function spaces with different number of variables,” Ukrainian Math. J. 50 (8), 1170–1191 (1998).
D. M. Bushev and Yu. I. Kharkevych, “Conditions of convergence almost everywhere for the convolution of a function with delta-shaped kernel to this function,” UkrainianMath. J. 67 (11), 1643–1661 (2015).
É. L. Shtark, “The complete asymptotic expansion for the measure of approximation of Abel-Poisson’s singular integral for Lip 1,” Mat. Zametki 13 (1), 21–28 (1973) [Math. Notes 13 (1), 14–16 (1973)].
L. P. Falaleev, “Approximation of conjugate functions by generalized Abel-Poisson operators,” Mat. Zametki 67 (4), 595–602 (2000) [Math. Notes 67 (4), 505–511 (2000)].
V. A. Baskakov, “Some properties of operators of Abel-Poisson type,” Mat. Zametki 17 (2), 169–180 (1975) [Math. Notes 17 (2), 101–107 (1975)].
Yu. I. Kharkevich and T. A. Stepanyuk, “Approximation properties of Poisson integrals for the classesCφ β Ha,” Mat. Zametki 96 (6), 939–952 (2014) [Math. Notes 96 (6), 1008–1019 (2014)].
L. P. Falaleev, “On approximation of functions by generalized Abel–Poisson operators,” Sibirsk. Mat. Zh. 42 (4), 926–936 (2001) [Sib.Math. J. 42 (4), 779–788 (2001)].
Yu. I. Kharkevych and I. V. Kal’chuk, “Approximation of (φ, β)-differentiable functions by Weierstrass integrals,” UkrainianMath. J. 59 (7), 1059–1087 (2007).
G. M. Polozhii, Equations of Mathematical Physics (Radyan’ska Shkola, Kiev, 1959) [in Ukranian].
S. Bochner, Lectures on Fourier Integrals (Princeton University Press, Princeton, NJ, 1959; Fizmatgiz, Moscow, 1962).
V. S. Vladimirov, Equations of Mathematical Physics (Nauka, Moscow, 1988) [in Russian].
A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics (Nauka, Moscow, 1966) [in Russian].
J. Garnett, Bounded Analytic Functions (Academic Press, New York, 1981; Mir, Moscow, 1984).
I. P. Natanson, Theory of Functions of a Real Variable (Nauka, Moscow, 1974) [in Russian].
Introduction to Fourier Analysis on Euclidean Spaces (Princeton Univ. Press, Princeton, NJ, 1971; Mir, Moscow, 1974).
S. Banach, A Course in Functional Analysis (Radyan’ska Shkola, Kiev, 1948) [in Ukrainian].
N. I. Akhiezer, Lectures in Approximation Theory (Nauka, Moscow, 1965) [in Russian].
N. P. Korneichuk, Exact Constants in Approximation Theory (Nauka, Moscow, 1987) [in Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © D. N. Bushev, Yu. I. Kharkevich, 2018, published in Matematicheskie Zametki, 2018, Vol. 103, No. 6, pp. 803–816.
Rights and permissions
About this article
Cite this article
Bushev, D.N., Kharkevich, Y.I. Finding Solution Subspaces of the Laplace and Heat Equations Isometric to Spaces of Real Functions, and Some of Their Applications. Math Notes 103, 869–880 (2018). https://doi.org/10.1134/S0001434618050231
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434618050231