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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Decay of Walsh series and dyadic differentiation


Author: William R. Wade
Journal: Trans. Amer. Math. Soc. 277 (1983), 413-420
MSC: Primary 42C10; Secondary 43A75
MathSciNet review: 690060
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Abstract: Let $ {W_2}\,n\,[f]$ denote the $ {2^n}{\text{th}}$ partial sums of the Walsh-Fourier series of an integrable function $ f$. Let $ {\rho _n}(x)$ represent the ratio $ {W_2}n[f,x]/{2^n}$, for $ x \in [0,1]$, and let $ T(f)$ represent the function $ {(\Sigma \rho _n^2)^{1/2}}$. We prove that $ T(f)$ belongs to $ {L^p}[0,1]$ for all $ 0 < p < \infty$. We observe, using inequalities of Paley and Sunouchi, that the operator $ f \to 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 [0, 1], 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_0^1 {\Sigma _{N = 1}^\infty \vert{W_N}[f,x] - {\sigma _N}[f,x]/N\,dx < \infty} $.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1983-0690060-X
PII: S 0002-9947(1983)0690060-X
Keywords: Walsh series, dyadic derivative, square function, maximal function, dyadic $ {H^1}$
Article copyright: © Copyright 1983 American Mathematical Society