On the integrability and L1-convergence of Walsh series with coefficients of bounded variation

Abstract

We study the integrability and the convergence in the L 1(0, 1)-metric of Walsh series ∑ k = 0 ∞ a k w k ( x), where { a k } is a null sequence of bounded variation. Our basic tool is the following Sidon type inequality: for every 1 < p ⩽ 2, sequence { a k } of real numbers, and integer n ⩾ 1, we have ∫ 0 1 ∑ k=1 n a kD k(x) dx⩽C pn 1− 1 p ∑ k=1 n |a k| p 1 p where D k ( x) = ∑ j = 0 k − 1 W j ( x) is the Dirichlet kernel for the Walsh system and C p = 2 p (p −) .