## Tauberian $L^{1}$-convergence classes of Fourier series. I

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- by William O. Bray and Časlav V. Stanojević
- Trans. Amer. Math. Soc.
**275**(1983), 59-69 - DOI: https://doi.org/10.1090/S0002-9947-1983-0678336-3
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## 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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## Bibliographic Information

- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**275**(1983), 59-69 - MSC: Primary 42A32; Secondary 42A20
- DOI: https://doi.org/10.1090/S0002-9947-1983-0678336-3
- MathSciNet review: 678336