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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A class of $ L\sp{1}$-convergence


Authors: R. Bojanić and Č. V. Stanojević
Journal: Trans. Amer. Math. Soc. 269 (1982), 677-683
MSC: Primary 42A16; Secondary 42A20, 42A32
MathSciNet review: 637717
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Abstract: It is proved that if the Fourier coefficients $ \{ {a_n}\} $ of $ f \in {L^1}(0,\,\pi )$ satisfy $ ({\ast}){n^{ - 1}}\sum\nolimits_{k = n}^{2n} {{k^p}\vert\Delta {a_n}\vert p = o(1)} $, for some $ 1 < p \leqslant 2$, then $ \vert\vert{s_n} - f\vert\vert = o(1)$, if and only if $ {a_n}\lg n = o(1)$. For cosine trigonometric series with coefficients of bounded variation and satisfying $ ({\ast})$ it is proved that a necessary and sufficient condition for the series to be a Fourier series is $ \{ {a_n}\} \in \mathcal{C}$, where $ \mathcal{C}$ is the Garrett-Stanojević [4] class.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1982-0637717-3
PII: S 0002-9947(1982)0637717-3
Keywords: $ {L^1}$-convergence of Fourier series, integrability of cosine series
Article copyright: © Copyright 1982 American Mathematical Society