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Proceedings of the American Mathematical Society

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



On the integrability and $ L\sp 1$-convergence of complex trigonometric series

Author: Ferenc Móricz
Journal: Proc. Amer. Math. Soc. 113 (1991), 53-64
MSC: Primary 42A10; Secondary 42A32
MathSciNet review: 1072345
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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

$\displaystyle \sum\limits_{m = 1}^\infty {{2^{m/q}}} {\left( {\sum\limits_{k = ...} \right\vert}^p}} } \right)^{1/p}} < \infty ,\frac{1}{p} + \frac{1}{q} = 1,$

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 _{\vert k\vert \to \infty }}{c_k}\ln \vert k\vert = 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].

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Keywords: Complex trigonometric series, null sequences of bounded variation, weakly even sequences, symmetric partial sums, modified trigonometric sums, pointwise convergence, Lebesgue integrability, convergence in $ {L^1}( - \pi ,\pi )$-norm, Sidon type inequalities, Bernstein's inequality
Article copyright: © Copyright 1991 American Mathematical Society