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On weighted integrability of trigonometric series and $ L\sp 1$-convergence of Fourier series

Authors: William O. Bray and Časlav V. Stanojević
Journal: Proc. Amer. Math. Soc. 96 (1986), 53-61
MSC: Primary 42A20
MathSciNet review: 813809
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Abstract: A result concerning integrability of $ f(x)L(1/x)(g(x)L(1/x))$, where $ f(x)(g(x))$ is the pointwise limit of certain cosine (sine) series and $ L( \cdot )$ is slowly vary in the sense of Karamata [5] is proved. Our result is an excluded case in more classical results (see [4]) and also generalizes a result of G. A. Fomin [1]. Also a result of Fomin and Telyakovskii [6] concerning $ {L^1}$-convergence of Fourier series is generalized. Both theorems make use of a generalized notion of quasi-monotone sequences.

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Keywords: Integrability of trigonometric series, slowly varying functions, regularly varying sequences, $ {L^1}$-convergence of Fourier series
Article copyright: © Copyright 1986 American Mathematical Society

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