Trigonometric series with noninteger harmonics

Abstract

Let $\{c_k\}$ be a nonincreasing sequence of positive numbers (more general classes of sequences are also considered), and $\alpha >0$ be not an integer. We find necessary and sufficient conditions for the uniform convergence of the series ${\sum }_k{c}_k\sin k^{\alpha }x$ and ${\sum }_k{c}_k\cos k^{\alpha }x$ on the real line and its bounded subsets.