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Tauberian $ L\sp{1}$-convergence classes of Fourier series. I


Authors: William O. Bray and Časlav V. Stanojević
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
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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 }\vert\Delta \hat{f}(n)\vert = 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

$\displaystyle l_n^{ - 1/q}{\left({\sum\limits_{k = n}^{n + [n/{l_n}]} {{k^{p - ... ...} {\vert^p}} \right)^{1/p}} = o(1)\, \quad (n \to \infty), \quad 1/p + 1/q = 1,$

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)\} $.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0678336-3
Keywords: $ {L^1}$-convergence of Fourier series
Article copyright: © Copyright 1983 American Mathematical Society

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