Yu, Dansheng and Zhou, Ping and Zhou, Songping (2007) *On Lp Integrability and Convergence of Trigonometric Series.* [Preprint]

## Abstract

We first give the necessary and sufficient condition for x^{-\gamma} \theta(x) \in L^p, 1 < p < \infty, 1/p−1 < \gamma < 1/p, where \theta(x) is the sum of either \sum_{k=1}^\infty a_k \cos kx or \sum_{k=1}^\infty b_k \sin kx and its associated Fourier coefficients are \lambda_n, n = 1, 2, ..., (i.e., \lambda_n is either a_n or b_n), under the condition that {\lambda_n} belongs to the class of so called Mean Value Bounded Variation Sequences (MVBVS). Then we discuss the relations among the Fourier coefficients \lambda_n and the sum function \theta(x), under the condition that {\lambda_n} \in MVBVS, and deduce a sharp estimate for the weighted modulus of continuity of \theta(x) in L^p norm.

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