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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the integrability and $L^ 1$-convergence of complex trigonometric series
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by Ferenc Móricz
Proc. Amer. Math. Soc. 113 (1991), 53-64
DOI: https://doi.org/10.1090/S0002-9939-1991-1072345-0

Abstract:

We prove that if a weakly even sequence $\{ {c_k}:k = 0, \pm 1, \ldots \}$ of complex numbers is such that for some $p > 1$ we have \[ \sum \limits _{m = 1}^\infty {{2^{m/q}}} {\left ( {\sum \limits _{k = {2^{m - 1}}}^{{2^m} - 1} {{{\left | {\Delta \left ( {{c_k} + {c_{ - k}}} \right )} \right |}^p}} } \right )^{1/p}} < \infty ,\frac {1}{p} + \frac {1}{q} = 1,\] then the symmetric partial sums of the trigonometric series $( * )\sum \nolimits _{k = - \infty }^\infty {{c_k}{e^{ikx}}}$ converge pointwise, except possibly at $x = 0(\operatorname {mod} 2\pi )$, to a Lebesgue integrable function, $( * )$ is the Fourier series of its sum, and series $( * )$ converges in ${L^1}( - \pi ,\pi )$-norm if and only if ${\lim _{|k| \to \infty }}{c_k}\ln |k| = 0$. In addition, we present new proofs of the theorems by J. Fournier and W. Self [6] and by ČC. V. Stanojević and V. B. Stanojević [10].
References
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Bibliographic Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 113 (1991), 53-64
  • MSC: Primary 42A10; Secondary 42A32
  • DOI: https://doi.org/10.1090/S0002-9939-1991-1072345-0
  • MathSciNet review: 1072345