Geodesic
Conditions under Which the Sums of Absolute Values of Blocks in the Fourier--Walsh Series for Functions of~Bounded Variation Belong to Spaces~$L^p$
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 28 (2022) no. 4, pp. 226-236

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In this paper, the following question is considered: what conditions on a strictly increasing sequence of positive integers $\{n_j\}_{j=1}^{\infty}$ guarantee that the sum of the series $$ \sum_{j=1}^{\infty}\bigg|\sum_{k=n_j}^{n_{j+1}-1}c_k(f) w_k(x)\bigg|,$$ where $c_k(f)$ are the Walsh–Fourier coefficients of a function $f$, belongs to the space $L^p[0,1)$, $p>1$, for any function $f$ of bounded variation? For $p=\infty$, it is proved that such a sequence does not exist. For finite $p>1$, sufficient conditions are obtained for the sequence $\{n_{j}\}$; these conditions are similar to the ones obtained by the first author in the trigonometric case.
Keywords: Walsh–Fourier series, functions of bounded variation
Mots-clés : $L^p$-spaces. 
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     author = {S. A. Telyakovskii and N. N. Kholshchevnikova},
     title = {Conditions under {Which} the {Sums} of {Absolute} {Values} of {Blocks} in the {Fourier--Walsh} {Series} for {Functions} {of~Bounded} {Variation} {Belong} to {Spaces~}$L^p$},
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S. A. Telyakovskii; N. N. Kholshchevnikova. Conditions under Which the Sums of Absolute Values of Blocks in the Fourier--Walsh Series for Functions of~Bounded Variation Belong to Spaces~$L^p$. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 28 (2022) no. 4, pp. 226-236. http://geodesic.mathdoc.fr/item/TIMM_2022_28_4_a20/