S.A. Telyakovskii, N.N. Kholshchevnikova. Conditions under which the sums of absolute values of blocks in the Fourier–Walsh series of functions of bounded variation belong to spaces $L^p$ ... P. 226-236

In this paper, the following question is considered: under what conditions on a strictly increasing sequence of natural numbers $\{n_j\}_{j=1}^{\infty}$ does 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 $f$, belong to the space $L^p[0,1)$, $p>1$, for any function $f$ of bounded variation? For the case $p=\infty$, 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, $L^p$-spaces

Received June 4, 2022

Revised September 23, 2022

Accepted September 26, 2022

Sergei Alexandrovich Telyakovskii$^†$, Dr. Phys.-Math. Sci., Prof., Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, 119991 Russia.

Natalia Nikolaevna Kholshchevnikova, Dr. Phys.-Math. Sci., Prof., Moscow State University of Technology “STANKIN”, Moscow, 127055 Russia, e-mail: kholshchevnikova@gmail.com

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Cite this article as: S.A. Telyakovskii, N.N. Kholshchevnikova. Conditions under which the sums of absolute values of blocks in the Fourier–Walsh series of functions of bounded variation belong to spaces $L^p$. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 4, pp. 226–236; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2022, Vol. 319, Suppl. 1, pp. S271–S280.