Uniform convergence criterion for non-harmonic sine series

TL;DR

This paper establishes uniform convergence criteria for non-harmonic sine series with monotone coefficients, depending on the power $\alpha$, and extends results to polynomial modifications and more general coefficient classes.

Contribution

It provides new uniform convergence conditions for sine series with non-harmonic frequencies, including polynomial modifications and broader coefficient classes.

Findings

For $\alpha ext{ in } (0,2)$, monotone $c_k$ with $c_k k o 0$ ensure uniform convergence.

When $\alpha$ is odd, the same condition guarantees convergence on all of $\mathbb{R}$.

For even $\alpha$, absolute summability of $c_k$ is necessary at specific points.

Abstract

We show that for a nonnegative monotone sequence $\{c_k\}$ the condition $c_kk\to 0$ is sufficient for uniform convergence of the series $\sum_{k=1}^{\infty}c_k\sin k^{\alpha} x$ on any bounded set for $\alpha\in (0,2)$, and for an odd natural $\alpha$ it is sufficient for uniform convergence on the whole $\mathbb{R}$. Moreover, the latter assertion still holds if we replace $k^{\alpha}$ by any polynomial in odd powers with rational coefficients. On the other hand, in the case of an even $\alpha$ it is necessary that $\sum_{k=1}^{\infty}c_k<\infty$ for convergence of the mentioned series at the point $\pi/2$ or at the point $2\pi/3$. Consequently, we obtain uniform convergence criteria. Besides, the results for a natural $\alpha$ remain true for sequences from more general RBVS class.