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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On moduli of continuity and divergence of Fourier series on groups.


Author: C. W. Onneweer
Journal: Proc. Amer. Math. Soc. 29 (1971), 109-112
MSC: Primary 42.50
MathSciNet review: 0287249
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Abstract: Let G be a 0-dimensional, metrizable, compact, abelian group. Then its character group X is a countable, discrete, torsion, abelian group. N. Ja. Vilenkin defined an enumeration for the elements of X and developed part of the Fourier theory on G. Among other things he proved on G a theorem similar to the Dini-Lipschitz test for trigonometric Fourier series. In this note we will show that Vilenkin's result is in some sense the best possible by proving the existence of a continuous function f on G whose modulus of continuity, $ {\theta _k}(f)$, satisfies $ {\theta _k}(f) = O({k^{ - 1}})$ as $ k \to \infty $ and whose Fourier series diverges at a point of G. The function f will be defined by means of the analogue in X of the classical Fejér polynomials.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0287249-8
PII: S 0002-9939(1971)0287249-8
Keywords: 0-dimensional group, metrizable group, compact abelian group, Fourier series, Walsh functions, modulus of continuity, Dini-Lipschitz test, Fejér polynomials
Article copyright: © Copyright 1971 American Mathematical Society