Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

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Tauberian $L^{1}$-convergence classes of Fourier series. IHTML articles powered by AMS MathViewer

by William O. Bray and Časlav V. Stanojević
Trans. Amer. Math. Soc. 275 (1983), 59-69 Request permission

Abstract:

It is shown that the Stanojević [2] necessary and sufficient conditions for ${L^1}$-convergence of Fourier series of $f \in {L^1}(T)$ can be reduced to the classical form. A number of corollaries of a recent Tauberian theorem are obtained for the subclasses of the class of Fourier coefficients satisfying ${n^\alpha }|\Delta \hat {f}(n)| = o(l) (n \to \infty )$ for some $0 < \alpha \leqslant \frac {1}{2}$. For Fourier series with coefficients asymptotically even with respect to a sequence $\{{l_n}\} ,{l_n} = o(n) (n \to \infty )$, and satisfying $l_n^{ - 1/q}{\left ({\sum \limits _{k = n}^{n + [n/{l_n}]} {{k^{p - 1}}|\Delta \hat f(k)} {|^p}} \right )^{1/p}} = o(1) \quad (n \to \infty ), \quad 1/p + 1/q = 1,$ necessary and sufficient conditions for ${L^1}$-convergence are obtained. In particular for ${l_n} = [\parallel {\sigma _n}(f) - f{\parallel ^{ - 1}}]$, an important corollary is obtained which connects smoothness of $f$ with smoothness of $\{\hat f(n)\}$.
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