  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.
