We first give a necessary and sufficient condition for , 1 < p < ∞, 1/p - 1 < γ < 1/p, where ϕ(x) is the sum of either or , 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 norm.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm182-3-3,
author = {Dansheng Yu and Ping Zhou and Songping Zhou},
title = {On $L^{p}$ integrability and convergence of trigonometric series},
journal = {Studia Mathematica},
volume = {178},
year = {2007},
pages = {215-226},
zbl = {1133.42007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm182-3-3}
}
Dansheng Yu; Ping Zhou; Songping Zhou. On $L^{p}$ integrability and convergence of trigonometric series. Studia Mathematica, Tome 178 (2007) pp. 215-226. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm182-3-3/