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On uniform convergence for Walsh-Fourier series

1970; Mathematical Sciences Publishers; Volume: 34; Issue: 1 Linguagem: Inglês

10.2140/pjm.1970.34.117

1945-5844

Cornelius Onneweer,

Differential Equations and Boundary Problems

In 1940 R. Salem formulated a sufficient condition for a continuous and periodic function to have a trigonometric Fourier series which converges uniformly to the function.In this paper we will formulate a similar condition, which implies that the Walsh-Fourier series of such a function has this property.Furthermore we show that our result is stronger than certain classical results, and that it also implies the uniform convergence of the Walsh-Fourier series of certain classes of continuous functions of generalized bounded variation.The latter is analogous to results obtained by L. C. Young and R. Salem for trigonometric Fourier series.Let {φ n (x)} be the sequence of Rademacher functions, i.e., φ Q (χ) = +1 (O £ x < i-) , φ o (x) =φ o (χ + 1) = φ o (x) .ψn (x) = φ o (2 n x), (n = 1, 2, 3, •).In [3] R. E. A. C. Paley gave the following definition for the Walsh functions {ψ n (x)}: ψ o (x) = 1, and,