Absolute summability of Walsh-Fourier series

Narendrakumar Ramanlal Ladhawala

Vol. 65 (1976), No. 1, 103–108

DOI: 10.2140/pjm.1976.65.103

Abstract

We prove that for all f ∈ℋ¹, ∑ _k=1^∞(1∕k)|f(k)|≤ K∥f∥_ℋ1, where ℋ¹ is the Walsh function analogue of the classical Hardy-space and f(k) is the k-th Walsh-Fourier coefficient of f. We obtain this as a consequence of its dual result: given a sequence {a_k} of numbers such that a_k = O(1∕k), there exists a function h ∈ BMO with ĥ(k) = a_k.

We study the relation between our results and the theory of differentiation on the Walsh group, developed by Butzer and Wagner.

Mathematical Subject Classification

Primary: 42A56, 42A56

Milestones

Received: 2 February 1976

Published: 1 July 1976