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Mat. Zametki, 2020, Volume 108, Issue 2, Pages 236–251 (Mi mz12575)  

On the Fourier–Walsh Transform of Functions from Dyadic Dini–Lipschitz Classes on the Semiaxis

S. S. Platonov

Petrozavodsk State University

Abstract: Let $f(x)$ be a function belonging to the Lebesgue class $L^p({\mathbb R}_+)$ on the semiaxis ${\mathbb R}_+=[0,+\infty)$, $1\le p\le 2$, and let $\widehat{f}$ be the Fourier–Walsh transform of the function $f$. In this paper, we give the solution of the following problem: if the function $f$ belongs to the dyadic Dini–Lipschitz class $\operatorname{DLip}_\oplus(\alpha,\beta,p;{\mathbb R}_+)$, $\alpha>0$, $\beta\in{\mathbb R}$, then for what values of $r$ can we guarantee that $\widehat{f}$ belongs to $L^r({\mathbb R}_+)$? The result obtained is an analog of the classical Titchmarsh theorem on the Fourier transform of functions from Lipschitz classes on ${\mathbb R}$.

Keywords: dyadic harmonic analysis, Dini–Lipschitz classes, Fourier–Walsh transform.

DOI: https://doi.org/10.4213/mzm12575

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English version:
Mathematical Notes, 2020, 108:2, 229–242

Bibliographic databases:

UDC: 517.986.62
Received: 26.09.2019

Citation: S. S. Platonov, “On the Fourier–Walsh Transform of Functions from Dyadic Dini–Lipschitz Classes on the Semiaxis”, Mat. Zametki, 108:2 (2020), 236–251; Math. Notes, 108:2 (2020), 229–242

Citation in format AMSBIB
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