Power Weight Integrability for Sums of Moduli of Blocks from Trigonometric Series

Abstract

We study the following problem: find conditions on sequences {γ(r)}, {n_j}, and {v_j} under which, for any sequence {b_k} such that b_k → 0 and ${\sum}_{k=r}^\infty|b_k-b_{k+1}|\leq\gamma(r)$ for all r ∈ ℕ, the integral $\int_0^\pi {{U^p}(x)/{x^q}dx}$ converges, where p > 0, q ∈ [1 − p; 1), and $U(x):=\sum\nolimits_{j = 1}^\infty {|\sum\nolimits_{k = {n_j}}^{{v_j}} {{b_k}\sin kx|} }$. For γ(r) = B/r with B > 0, this problem was studied and solved by S. A.Telyakovskii. In the case where p ≥ 1, q = 0, v_j = n_j+1 − 1, and {b_k} is monotone, A. S. Belov obtained a criterion for the function U(x) to belong to the space L_p. In Theorem1 of the present paper, we give sufficient conditions for the convergence of the above integral, which coincide with the Telyakovskii sufficient conditions for γ(r) = B/r with B > 0. In the case γ(r) = O(1/r), the Telyakovskii conditions may be violated while 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 of necessary conditions for the convergence of the integral $\int_0^\pi {{U^p}(x)/{x^q}dx}$, where p > 0 and q ∈ [1 − p; 1), remains open.