Trigonometric series of Nikol’skii classes

Abstract

We study when sums of trigonometric series belong to given function classes. For this purpose we describe the Nikol'skii class of functions and, in particular, the generalized Lipschitz class. Results for series with positive and general monotone coefficients are presented.

On the degree of continuity and smoothness of sine and cosine Fourier transforms of Lebesgue integrable functions

“…We note that the analogous problem is solved in the special case of sine series with positive coefficients tending to zero, see [7] by Tikhonov.…”

“…Theorems 1-7 have their periodic analogues for sine and cosine series. We refer to the papers [1] by Boas, [6] by Németh, [7] by Tikhonov in the special case where the coefficients {a k : k 1} of the series are positive numbers such that ∞ k=1 a k < ∞; and to the papers [3,4] by the present author in the case where the coefficients {c k : k 1} of the series are complex numbers such that ∞ k=1 |c k | < ∞.…”

“…The author thanks the anonymous referee for careful reading of the paper and calling our attention to Reference [7].…”

Addendum to “Trigonometric series of Nikol’skii classes”

“…The following two theorems on trigonometric series from the Nikol'skii class W α H β (γ) are simple corollaries of Theorems 2.12.2 (see [6,Lemmas 2.112.12]). Theorem 4.1 (cosine).…”

“…The history of this question can be found in [6]. We also study a problem of embedding between the Nikol'skii class and the class of strong sums of Fourier series.…”