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On Grasia's criterion for uniform convergence of Fourier series

Part of: Harmonic analysis in one variable

Published online by Cambridge University Press:  09 April 2009

Charles Oehring
Charles Oehring
Affiliation:
Virginia Polytechnic Institute & State University Blacksburg, Virginia 24061-0123, U.S.A.
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Abstract

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Garsia's discovery that functions in the periodic Besov space λ(p-1, p, 1), with 1 < p < ∞, have uniformly convergent Fourier series prompted him, and others, to seek a proof based on one of the standard convergence tests. on one of the standard convergence tests. We show that Lebesgue's test is adequate, whereas Garsia's criterion is independent of other classical critiera (for example, that of Dini-Lipschitz). The method of proof also produces a sharp estimate for the rate of uniform convergence for functions in λ(p-1, p, 1). Further, it leads to a very simple proof of the embedding theorem for these spaces, which extends (though less simply) to λ(α, p, q)

MSC classification

Secondary: 42A20: Convergence and absolute convergence of Fourier and trigonometric series
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 51 , Issue 2 , October 1991 , pp. 305 - 323
Copyright
Copyright © Australian Mathematical Society 1991

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