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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Generalizations of the Sidon-Telyakovskiĭ theorem


Authors: Časlav V. Stanojević and Vera B. Stanojevic
Journal: Proc. Amer. Math. Soc. 101 (1987), 679-684
MSC: Primary 42A20; Secondary 42A32
MathSciNet review: 911032
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Abstract: The well-known Sidon-Telyakovskii integrability condition is considerably lightened as follows:

$\displaystyle \frac{1}{n}\sum\limits_{k = 1}^n {\frac{{\vert\Delta c(k){\vert^p}}}{{A_k^p}} = O(1),\quad n \to \infty } ,$

where $ \{ c(n)\} $ is a certain null-sequence and $ 1 < p \leq 2$. It is proved that $ \sum\nolimits_{n = 1}^\infty {{n^{p - 1}}\vert\Delta c(n){\vert^p}{\rho ^p}(n) < \infty } $ is also a sufficient integrability condition provided $ \sum\nolimits_{n = 1}^\infty {(1/n\rho (n)) < \infty } $, where $ \{ \rho (n)\} $ is an increasing sequence of positive numbers.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1987-0911032-2
PII: S 0002-9939(1987)0911032-2
Keywords: Integrability of trigonometric series, $ {L^1}$-convergence of Fourier series
Article copyright: © Copyright 1987 American Mathematical Society