The Hardy-Littlewood theorem for trigonometric series with generalized monotone coefficients

Abstract

Earlier we introduced a continuous scale of monotony for sequences (classes M _α, α ≥ 0), where, for example, M ₀ is the set of all nonnegative vanishing sequences, M ₁ is the class of all nonincreasing sequences, tending to zero, etc. In addition, we extended several results obtained for trigonometric series with monotone convex coefficients onto more general classes. The main result of this paper is a generalization of the well-known Hardy—Littlewood theorem for trigonometric series, whose coefficients belong to classes M _α, where α ∈ ($\tfrac{1} {2}$, 1). Namely, the following assertion is true.

Let α ∈ ($\tfrac{1} {2}$, 1), $\tfrac{1} {\alpha }$ < p < 2, a sequence a ∈ M _α, and $\sum\limits_{n = 1}^\infty {a_n^p n^{p - 2} } < \infty$. Then the series $\tfrac{{a_0 }} {2} + \sum\limits_{n = 1}^\infty {a_n }$ cos nx converges on (0,2π) to a finite function f(x) and f(x) ∈ L _p (0,2π).