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

Abstract

We study the following problem: find conditions on sequences {γ(r)}, {nj}, and {vj} under which, for any sequence {bk} such that bk → 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, vj = nj+1 − 1, and {bk} is monotone, A. S. Belov obtained a criterion for the function U(x) to belong to the space Lp. 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.