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 Trudy Inst. Mat. i Mekh. UrO RAN, 2017, Volume 23, Number 3, Pages 125–133 (Mi timm1443)

Power wight integrability for sums of moduli of blocks from trigonometric series

V. P. Zastavnyi, A. S. Levadnaya

Donetsk National University

Abstract: The following problem is studied: find conditions on sequences $\{\gamma(r)\}$, $\{n_j\}$, and $\{v_j\}$ under which, for any sequence $\{b_k\}$ such that $\sum_{k=r}^{\infty}|b_k-b_{k+1}|\leq\gamma(r)$, $b_k\to 0$, the integral $\int_0^\pi U^p(x)/{x^q}dx$ is convergent, where $p>0$, $q\in[1-p;1)$, and $U(x):=\sum_{j=1}^{\infty}|\sum_{k=n_j}^{v_j}b_k \sin kx|$. In the case $\gamma(r)={B}/{r}$, $B>0$, this problem was studied and solved by S. A. Telyakovskii. In the case where $p\ge 1$, $q=0$, $v_j=n_{j+1}-1$, and the sequence $\{b_k\}$ is monotone, A. S. Belov obtained a criterion for the belonging of the function $U(x)$ to the space $L_p$. In Theorem 1 of the present paper, we give sufficient conditions for the convergence of the above integral, which for $\gamma(r)= B/{r}$, $B>0$, coincide with Telyakovskii's sufficient conditions. In the case $\gamma(r)= O(1/{r})$, Telyakovskii's conditions may be violated, but the application of Theorem 1 guarantees the convergence of the integral. The corresponding examples are given in the last section of the paper. The question on necessary conditions for the convergence of the integral $\int_0^\pi U^p(x)/{x^q}dx$, where $p>0$ and $q\in[1-p;1)$, remains open.

Keywords: trigonometric series, sums of moduli of blocks, power weight.

DOI: https://doi.org/10.21538/0134-4889-2017-23-3-125-133

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Bibliographic databases:

UDC: 517.518.45
MSC: 42A32

Citation: V. P. Zastavnyi, A. S. Levadnaya, “Power wight integrability for sums of moduli of blocks from trigonometric series”, Trudy Inst. Mat. i Mekh. UrO RAN, 23, no. 3, 2017, 125–133

Citation in format AMSBIB
\Bibitem{ZasLev17} \by V.~P.~Zastavnyi, A.~S.~Levadnaya \paper Power wight integrability for sums of moduli of blocks from trigonometric series \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2017 \vol 23 \issue 3 \pages 125--133 \mathnet{http://mi.mathnet.ru/timm1443} \crossref{https://doi.org/10.21538/0134-4889-2017-23-3-125-133} \elib{http://elibrary.ru/item.asp?id=29938005}