J Austral Math Soc Ser A 51 pp305--323, 1991.

On Garsia's Criterion for Uniform Convergence of Fourier Series

Charles Oehring

Abstract

Garsia's discovery that functions in the periodic Besov space L(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. We show that Lesbesgue's test is adequate, whereas Garsia's criterion is independent of other classical criteria (for example, that of Dini-Lipschitz). The method of proof also produces a sharp estimate for the rate of uniform convergence for functions in L(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 L(a, p, q).

1980 AMS Subject Classification (1985 Revision): 42A20

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Authors

Charles Oehring

Virginia Polytechnic Institute & State University, Blacksburg, Virginia 24061-0123, U.S.A.