Geodesic
On the Fourier–Walsh Transform of Functions from Dyadic Dini–Lipschitz Classes on the Semiaxis
Matematičeskie zametki, Tome 108 (2020) no. 2, pp. 236-251

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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. 
S. S. Platonov. On the Fourier–Walsh Transform of Functions from Dyadic Dini–Lipschitz Classes on the Semiaxis. Matematičeskie zametki, Tome 108 (2020) no. 2, pp. 236-251. http://geodesic.mathdoc.fr/item/MZM_2020_108_2_a7/
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     title = {On the {Fourier{\textendash}Walsh} {Transform} of {Functions} from {Dyadic} {Dini{\textendash}Lipschitz} {Classes} on the {Semiaxis}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {236--251},
     year = {2020},
     volume = {108},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2020_108_2_a7/}
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