# On ${L}^{p}$ integrability and convergence of trigonometric series

Studia Mathematica (2007)

• Volume: 182, Issue: 3, page 215-226
• ISSN: 0039-3223

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## Abstract

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We first give a necessary and sufficient condition for ${x}^{-\gamma }\varphi \left(x\right)\in {L}^{p}$, 1 < p < ∞, 1/p - 1 < γ < 1/p, where ϕ(x) is the sum of either ${\sum }_{k=1}^{\infty }{a}_{k}coskx$ or ${\sum }_{k=1}^{\infty }{b}_{k}sinkx$, under the condition that λₙ (where λₙ is aₙ or bₙ respectively) belongs to the class of so called Mean Value Bounded Variation Sequences (MVBVS). Then we discuss the relations among the Fourier coefficients λₙ and the sum function ϕ(x) under the condition that λₙ ∈ MVBVS, and deduce a sharp estimate for the weighted modulus of continuity of ϕ(x) in ${L}^{p}$ norm.

## How to cite

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Dansheng Yu, Ping Zhou, and Songping Zhou. "On $L^{p}$ integrability and convergence of trigonometric series." Studia Mathematica 182.3 (2007): 215-226. <http://eudml.org/doc/284612>.

@article{DanshengYu2007,
abstract = {We first give a necessary and sufficient condition for $x^\{-γ\} ϕ(x) ∈ L^\{p\}$, 1 < p < ∞, 1/p - 1 < γ < 1/p, where ϕ(x) is the sum of either $∑_\{k=1\}^\{∞\} a_\{k\} cos kx$ or $∑_\{k=1\}^\{∞\} b_\{k\} sin kx$, under the condition that λₙ (where λₙ is aₙ or bₙ respectively) belongs to the class of so called Mean Value Bounded Variation Sequences (MVBVS). Then we discuss the relations among the Fourier coefficients λₙ and the sum function ϕ(x) under the condition that λₙ ∈ MVBVS, and deduce a sharp estimate for the weighted modulus of continuity of ϕ(x) in $L^\{p\}$ norm.},
author = {Dansheng Yu, Ping Zhou, Songping Zhou},
journal = {Studia Mathematica},
keywords = {Fourier series; modulus of continuity; mean value bounded variation sequences},
language = {eng},
number = {3},
pages = {215-226},
title = {On $L^\{p\}$ integrability and convergence of trigonometric series},
url = {http://eudml.org/doc/284612},
volume = {182},
year = {2007},
}

TY - JOUR
AU - Dansheng Yu
AU - Ping Zhou
AU - Songping Zhou
TI - On $L^{p}$ integrability and convergence of trigonometric series
JO - Studia Mathematica
PY - 2007
VL - 182
IS - 3
SP - 215
EP - 226
AB - We first give a necessary and sufficient condition for $x^{-γ} ϕ(x) ∈ L^{p}$, 1 < p < ∞, 1/p - 1 < γ < 1/p, where ϕ(x) is the sum of either $∑_{k=1}^{∞} a_{k} cos kx$ or $∑_{k=1}^{∞} b_{k} sin kx$, under the condition that λₙ (where λₙ is aₙ or bₙ respectively) belongs to the class of so called Mean Value Bounded Variation Sequences (MVBVS). Then we discuss the relations among the Fourier coefficients λₙ and the sum function ϕ(x) under the condition that λₙ ∈ MVBVS, and deduce a sharp estimate for the weighted modulus of continuity of ϕ(x) in $L^{p}$ norm.
LA - eng
KW - Fourier series; modulus of continuity; mean value bounded variation sequences
UR - http://eudml.org/doc/284612
ER -

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