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

Dansheng Yu; Ping Zhou; Songping Zhou

Studia Mathematica (2007)

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

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topDansheng 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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