Uniform Convergence of Trigonometric Series with Rarely Changing Coefficients

We consider the series $\sum _{k=1}^\infty a_k\sin kx$ and $\frac {a_0}2+\sum _{k=1}^\infty a_k\cos kx$ whose coefficients satisfy the condition $a_k=a_{n_m}$ for $n_{m-1}<k\le n_m$ , where the sequence $\{n_m\}$ can be expressed as the union of a finite number of lacunary sequences. The following results are obtained. If $ka_k\to0$ as $k\to\infty$, then the series $\sum _{k=1}^\infty a_k\sin kx$ is uniformly convergent. If $k|a_k|\le C$ for all $k$, then the sequence of partial sums of this series is uniformly bounded. If the series $\frac {a_0}2+\sum _{k=1}^\infty a_k\cos kx$ is convergent for $x=0$ and $ka_k\to0$ as $k\to\infty$, then this series is uniformly convergent. If the sequence of partial sums of the series $\frac {a_0}2+\sum _{k=1}^\infty a_k\cos kx$ for $x=0$ is bounded and $k|a_k|\le C$ for all $k$, then the sequence of partial sums of this series is uniformly bounded. In these assertions, conditions on the rates of decrease of the coefficients of the series are also necessary if the sequence $\{n_m\}$ is lacunary. In the general case, they are not necessary.