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• # Artykuł - szczegóły

## Studia Mathematica

2007 | 182 | 3 | 215-226

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

EN

### Abstrakty

EN
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.

215-226

wydano
2007

### Twórcy

autor
• Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China
• Department of Mathematics, Statistics and Computer Science, St. Francis Xavier University, Antigonish, Nova Scotia Canada, B2G 2W5
autor
• Department of Mathematics, Statistics and Computer Science, St. Francis Xavier University, Antigonish, Nova Scotia Canada, B2G 2W5
autor
• Institute of Mathematics, Zhejiang Sci-Tech University, Xiasha Economic Development Area, Hangzhou, Zhejiang 310018, China
• Department of Mathematics, Statistics and Computer Science, St. Francis Xavier University, Antigonish, Nova Scotia Canada, B2G 2W5