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Proceedings of the American Mathematical Society

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Convergence and integrability of trigonometric series with coefficients of bounded variation of order $ (m,p)$


Author: Vera B. Stanojevic
Journal: Proc. Amer. Math. Soc. 114 (1992), 711-718
MSC: Primary 42A32
DOI: https://doi.org/10.1090/S0002-9939-1992-1068132-0
MathSciNet review: 1068132
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Abstract: Let $ \{ c(n)\} $ be a complex null sequence such that for some integer $ m \geq 1$ and some $ p \in (1,2]$

$\displaystyle \sum\limits_{\vert n\vert < \infty } {\vert{\Delta ^m}c(n){\vert^... ...um\limits_{n = 1}^\infty {\vert\Delta (c(n) - c( - n))\vert\lg n < \infty .} } $

It is shown that the series

$\displaystyle ( * )\quad \sum\limits_{\vert n\vert < \infty } {c(n)} {e^{\operatorname{int} }},\quad t \in T = \frac{\mathbb{R}}{{2\pi \mathbb{Z}}}$

converges a.e. and that the well-known condition $ {C_w}$ of J. W. Garrett and C. V. Stanojevic [4, 3] implies that the series (*) is the Fourier series of its sum. This generalizes results of W. O. Bray and C. V. Stanojevic [1]. An important consequence of the main result is that $ n\Delta c(n) = 0(1),\quad \vert n\vert \to \infty $, implies that the condition $ {C_w}$ is equivalent to the de la Vallee Poussin summability of partial sums $ \{ {S_n}(c)\} $ as conjectured in [8].

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DOI: https://doi.org/10.1090/S0002-9939-1992-1068132-0
Keywords: Convergence and integrability of trigonometric series, sequences of bounded variation of order $ (m,p)$
Article copyright: © Copyright 1992 American Mathematical Society