( C ,α) Summability of Walsh—Fourier Series

Abstract

The one-dimensional dyadic martingale Hardy spaces Hp are introduced and it is proved that the maximal operator of the (C,α) means of a Walsh—Fourier series is bounded from Hp to Lp (1/(α + 1) < p < ∞) and is of weak type (L1,L1). As a consequence, we obtain the summability result due to Fine; more exactly, the (C,α) means of the Walsh—Fourier series of a function f ∈ L1 converge a.e. to the function in question. Moreover, we prove that the (C,α) means are uniformly bounded on Hp whenever 1/(α + 1) < p < ∞. We define the two-dimensional dyadic hybrid Hardy space H1♯ and verify that the maximal operator of the (C,α,β) means of a two-dimensional function is of weak type H1♯,L1). Consequence, the Walsh—Fourier series of every function f ∈ H1♯ is (C,α,β) summable to the function f.