Vol. 65, No. 1, 1976

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Absolute summability of Walsh-Fourier series

Narendrakumar Ramanlal Ladhawala

Vol. 65 (1976), No. 1, 103–108

We prove that for all f ∈ℋ1,  k=1(1∕k)|f(k)|≤ Kf1, where 1 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 {ak} of numbers such that ak = O(1∕k), there exists a function h BMO with ĥ(k) = ak.

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
Received: 2 February 1976
Published: 1 July 1976
Narendrakumar Ramanlal Ladhawala