Decay of Walsh Series and Dyadic Differentiation

William R. Wade
1983 Transactions of the American Mathematical Society  
Let W2n[f] denote the 2"th partial sums of the Walsh-Fourier series of an integrable function/. Let p"(x) represent the ratio W2n[f, x]/2n, for x e [0,1], and let T(f) represent the function (2p;j)'/2. We prove that T(f) belongs to //[0,1] for all 0 < p < oo. We observe, using inequalities of Paley and Sunouchi, that the operator/ -» T(f) arises naturally in connection with dyadic differentiation. Namely, if / is strongly dyadically differentiable (with derivative Df) and has average zero on
more » ... interval [0,1], then the Lp norms of /and T(Df) are equivalent when 1 <p < oo. We improve inequalities implicit in Sunouchi's work for the case p = 1 and indicate how they can be used to estimate the L1 norm of T(Df) and the dyadic //' norm of/by means of mixed norms of certain random Walsh series. An application of these estimates establishes that if /is strongly dyadically differentiable in dyadic H\ then /<j255=i I WN[f, x] -oN[f, x]/Ndx < oo.
doi:10.2307/1999364 fatcat:o2vv54agyvabhe5vlzb5cga5oe