ON THE INTEGRABILITY AND L1-CONVERGENCE OF WALSH-SERIES WITH COEFFICIENTS OF BOUNDED VARIATION

We study the integrability and the convergence in the L1(0, 1)-metric of Walsh series ∑k = 0∞akwk(x), where {ak} is a null sequence of bounded variation. Our basic tool is the following Sidon type inequality: for every 1 < p ≤ 2, sequence {ak} of real numbers, and integer n ≥ 1, we have ∫ 0 1 ∑ k=1 nakDk(x)dx≤Cpn1- 1 p ∑ k=1 n|ak|p 1 p where Dk(x) = ∑j = 0k - 1Wj(x) is the Dirichlet kernel for the Walsh system and Cp = 2 p (p -). © 1990.