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Decay of Walsh Series and Dyadic Differentiation

1983
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Transactions of the American Mathematical Society
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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

doi:10.2307/1999364
fatcat:o2vv54agyvabhe5vlzb5cga5oe