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