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# Decay of Walsh Series and Dyadic Differentiation

Transactions of the American Mathematical Society
Vol. 277, No. 1 (May, 1983), pp. 413-420
DOI: 10.2307/1999364
Stable URL: http://www.jstor.org/stable/1999364
Page Count: 8
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## Abstract

Let $W_2n\lbrack f \rbrack$ denote the $2^n$th partial sums of the Walsh-Fourier series of an integrable function $f$. Let $\rho_n(x)$ represent the ratio $W_2n\lbrack f, x \rbrack/2^n$, for $x \in \lbrack 0, 1 \rbrack$, and let $T(f)$ represent the function $(\sum\rho^2_n)^{1/2}$. We prove that $T(f)$ belongs to $L^p\lbrack 0, 1 \rbrack$ for all $0 < p , \infty$. We observe, using inequalities of Paley and Sunouchi, that the operator $f \rightarrow T(f)$ arises naturally in connection with dyadic differentiation. Namely, if $f$ is strongly dyadically differentiable (with derivative $\dot Df$) and has average zero on the interval $\lbrack 0, 1 \rbrack$, then the $L^p$ norms of $f$ and $T(\dot Df)$ are equivalent when $1 < p < \infty$. We improve inequalities implicit in Sunouchi's work for the case $p = 1$ and indicate how they can be used to estimate the $L^1$ norm of $T(\dot Df)$ and the dyadic $H^1$ norm of $f$ by means of mixed norms of certain random Walsh series. An application of these estimates establishes that if $f$ is strongly dyadically differentiable in dyadic $H^1$, then $\int^1_0\sum^\infty_{N=1}|W_N\lbrack f, x \rbrack - \sigma_N \lbrack f, x \rbrack/N dx < \infty$.

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