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Title: Two-dimensional Waterman classes and u-convergence of Fourier series

Abstract

New results on the u-convergence of the double Fourier series of functions from Waterman classes are obtained. It turns out that none of the Waterman classes wider than BV(T{sup 2}) ensures even the uniform boundedness of the u-sums of the double Fourier series of functions in this class. On the other hand, the concept of u(K)-convergence is introduced (the sums are taken over regions that are forbidden to stretch along coordinate axes) and it is proved that for functions f(x,y) belonging to the class {lambda}{sub 1/2}BV(T{sup 2}), where {lambda}{sub a}={l_brace}n{sup 1/2}/(ln(n+1)){sup a}{r_brace}{sub n=1}{sup {infinity}}, the corresponding u(K)-partial sums are uniformly bounded, while if f(x,y) element of {lambda}{sub a}BV(T{sup 2}), where a<1/2, then the double Fourier series of f(x,y) is u(K)-convergent everywhere.

Authors:
 [1]
  1. M.V. Lomonosov Moscow State University, Moscow (Russian Federation)
Publication Date:
OSTI Identifier:
21202881
Resource Type:
Journal Article
Journal Name:
Sbornik. Mathematics
Additional Journal Information:
Journal Volume: 190; Journal Issue: 7; Other Information: DOI: 10.1070/SM1999v190n07ABEH000414; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1064-5616
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CONVERGENCE; COORDINATES; FUNCTIONS; TWO-DIMENSIONAL CALCULATIONS

Citation Formats

D'yachenko, M I. Two-dimensional Waterman classes and u-convergence of Fourier series. United States: N. p., 1999. Web. doi:10.1070/SM1999V190N07ABEH000414; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
D'yachenko, M I. Two-dimensional Waterman classes and u-convergence of Fourier series. United States. doi:10.1070/SM1999V190N07ABEH000414; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
D'yachenko, M I. Tue . "Two-dimensional Waterman classes and u-convergence of Fourier series". United States. doi:10.1070/SM1999V190N07ABEH000414; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
@article{osti_21202881,
title = {Two-dimensional Waterman classes and u-convergence of Fourier series},
author = {D'yachenko, M I},
abstractNote = {New results on the u-convergence of the double Fourier series of functions from Waterman classes are obtained. It turns out that none of the Waterman classes wider than BV(T{sup 2}) ensures even the uniform boundedness of the u-sums of the double Fourier series of functions in this class. On the other hand, the concept of u(K)-convergence is introduced (the sums are taken over regions that are forbidden to stretch along coordinate axes) and it is proved that for functions f(x,y) belonging to the class {lambda}{sub 1/2}BV(T{sup 2}), where {lambda}{sub a}={l_brace}n{sup 1/2}/(ln(n+1)){sup a}{r_brace}{sub n=1}{sup {infinity}}, the corresponding u(K)-partial sums are uniformly bounded, while if f(x,y) element of {lambda}{sub a}BV(T{sup 2}), where a<1/2, then the double Fourier series of f(x,y) is u(K)-convergent everywhere.},
doi = {10.1070/SM1999V190N07ABEH000414; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)},
journal = {Sbornik. Mathematics},
issn = {1064-5616},
number = 7,
volume = 190,
place = {United States},
year = {1999},
month = {8}
}