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

V. P. Zastavnyi ; A. S. Levadnaya

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 3, pp. 125-133

Résumé

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 $U(x):=\sum_{j=1}^{\infty}\left|\sum_{k=n_j}^{v_j}b_k \sin kx\right|$. 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.

Détail

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

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