Trigonometric series with monotone coefficients

Abstract

Let {αn} be a monotonically decreasing sequence. Then each sequence {bn} such that bn ↓ 0, bn⩽αn, n=1, 2,..., is a sequence of Fourier-Lebesgue coefficients with respect to the system {cos nx} if and only if the sequence\(\sum\nolimits_{n = 1}^\infty {\frac{{{}^an}}{n}} \) converges.