Absolutely convergent Fourier series and function classes

[摘要] We study the smoothness property of a function f with absolutely convergent Fourier series, and give best possible sufficient conditions in terms of its Fourier coefficients to ensure that f belongs either to one of the Lipschitz classes Lip(alpha) and lip(alpha) for some 0 < alpha <= 1, or to one of the Zygmund classes Lambda(*)(1) and Our theorems generalize some of those by Boas [R.P. Boas Jr., Fourier series with positive coefficients, J. Math. Anal. Appl. 17 (1967) 463-483] and one by Nemeth [J. Nemeth, Fourier series with positive coefficients and generalized Lipschitz classes, Acta Sci. Math. (Szeged) 54 (1990) 291-304]. We also prove a localized version of a theorem by Paley [R.E.A.C. Paley, On Fourier series with positive coefficients, J. London Math. Soc. 7 (1932) 205-208] on the existence and continuity of the derivative of f. (c) 2006 Elsevier Inc. All rights reserved.

[发布日期] 2006-12-15

[关键词] absolutely convergent Fourier series;Lipschitz classes Lip(alpha) and lip(alpha);Zygmund classes A(*)(alpha) and Lambda(*)(alpha);formal differentiation of Fourier series