Trigonometric series with monotone coefficients

Let {_n} be a monotonically decreasing sequence. Then each sequence {b_n} such that b_n 0, b_nn, n=1, 2,..., is a sequence of Fourier-Lebesgue coefficients with respect to the system {cos nx} if and only if the sequence converges.Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 77–83, July, 1977.