Uniform convergence criterion for non-harmonic sine series

Abstract

Abstract We show that for a nonnegative monotonic sequence the condition is sufficient for the series to converge uniformly on any bounded set for , and for any odd it is sufficient for it to converge uniformly on the whole of . Moreover, the latter assertion still holds if we replace by any polynomial in odd powers with rational coefficients. On the other hand, in the case of even it is necessary that for the above series to converge at the point or at . As a consequence, we obtain uniform convergence criteria. Furthermore, the results for natural numbers remain true for sequences in the more general class . Bibliography: 17 titles.