ON GARSIA CRITERION FOR UNIFORM-CONVERGENCE OF FOURIER-SERIES

Garsia's discovery that functions in the periodic Besov space LAMBDA-(p-1, p, 1), with 1 < p < infinity, have uniformly convergent Fourier series prompted him, and others, to seek a proof based 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 LAMBDA-(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 LAMBDA-(alpha, p, q).