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On L 1 -convergence of Walsh-Fourier series

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  • C. W. Onneweer

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 L 1 ( G ) which implies that their Walsh-Fourier series converges in L 1 ( G ) -norm. As a corollary we obtain a Dini-Lipschitz-type theorem for L 1 ( G ) convergence and we prove that the assumption on the L 1 ( 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.

Suggested Citation

  • C. W. Onneweer, 1978. "On L 1 -convergence of Walsh-Fourier series," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 1, pages 1-10, January.
    • Handle: RePEc:hin:jijmms:373819
    DOI: 10.1155/S016117127800006X
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