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

On a theorem of P. L. Uljanov


Author: Vera B. Stanojevic
Journal: Proc. Amer. Math. Soc. 90 (1984), 370-372
MSC: Primary 42A20; Secondary 42A32
MathSciNet review: 728350
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Abstract: It is shown that if $ c\left( n \right) = o\left( 1 \right)$, $ \left\vert n \right\vert \to \infty $, and $ {\sum _{\left\vert n \right\vert}}_{ < \infty }\left\vert {{\Delta ^m}c\left( n \right)} \right\vert < \infty $, for some integer $ m \geqslant 1$, then the series $ {\sum _{\left\vert n \right\vert}}_{ < \infty }c\left( n \right){e^{\operatorname{int} }}$ converges to some $ f \in {L^p}\left( {\mathbf{T}} \right)$ for any $ 0 < p < 1 / m$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1984-0728350-0
PII: S 0002-9939(1984)0728350-0
Keywords: Convergence in $ {L^P}\left( {\mathbf{T}} \right)$-metric $ \left( {0 < p < 1} \right)$ of trigonometric series
Article copyright: © Copyright 1984 American Mathematical Society