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

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Yu D., Zhou P., Zhou S.

Studia Mathematica

2007

tom 182

nr 3

215-226

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.

Ograniczanie wyników

1 Studia Mathematica

1 Yu D.

1 Zhou P.

1 Zhou S.

1 2007

Wyniki wyszukiwania

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