International Journal of Mathematics and Mathematical Sciences
Volume 1 (1978), Issue 1, Pages 47-56
doi:10.1155/S016117127800006X
On L1-convergence of Walsh-Fourier series
Department of Mathematics, University of New Mexico, Albuquerque 87131, New Mexico, USA
Received 14 February 1977; Revised 25 October 1977
Copyright © 1978 C. W. Onneweer. 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.
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.