International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 1978 / Article

Open Access

Volume 1 |Article ID 373819 | https://doi.org/10.1155/S016117127800006X

C. W. Onneweer, "On L1-convergence of Walsh-Fourier series", International Journal of Mathematics and Mathematical Sciences, vol. 1, Article ID 373819, 10 pages, 1978. https://doi.org/10.1155/S016117127800006X

On L1-convergence of Walsh-Fourier series

Received14 Feb 1977
Revised25 Oct 1977

Abstract

Let G denote the dyadic group, which has as its dual group the Walsh(-Paley) functions. In this paper we formulate a condition for functions in L1(G) which implies that their Walsh-Fourier series converges in L1(G)-norm. As a corollary we obtain a Dini-Lipschitz-type theorem for L1(G) convergence and we prove that the assumption on the L1(G) modulus of continuity in this theorem cannot be weakened. Similar results also hold for functions on the circle group T and their (trigonometric) Fourier series.

Copyright © 1978 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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