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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Absolute convergence of series of Fourier coefficients


Author: James R. McLaughlin
Journal: Trans. Amer. Math. Soc. 184 (1973), 291-316
MSC: Primary 42A28; Secondary 42A56
DOI: https://doi.org/10.1090/S0002-9947-1973-0336203-2
MathSciNet review: 0336203
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Abstract: In this article the author unifies and generalizes practically all known sufficiency results for absolute convergence of series of Fourier coefficients that are given in terms of the integrated modulus of continuity, best approximation, or bounded pth variation. This is done for the trigonometric, Walsh, Haar, Franklin, and related systems as well as general orthonormal systems. Many of the original proofs of previous results relied upon special properties of the trigonometric, Haar, and other systems and were done independently of one another. Also, several authors have proved results which at the time they believed to be generalizations of past results, but are, in fact, corollaries of them. The present author will expose underlying principles and illustrate their usefulness.


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DOI: https://doi.org/10.1090/S0002-9947-1973-0336203-2
Keywords: Integrated modulus of continuity, best approximation, bounded pth variation, Lipschitz conditions, quasi-monotone sequences, trigonometric system, Walsh system, generalized Walsh system, Haar system, Franklin system, generalized Haar system, general orthogonal series, basis in $ {L^p}$
Article copyright: © Copyright 1973 American Mathematical Society