On the convergence (upper boundness) of trigonometric series

In this paper we prove that the condition 2n ∑ k=[ 2 ] kλk |n− k|+ 1 = o(1) (= O(1)) , for r = 0, 1, 2, . . . , is necessary for the convergence of the r − th derivative of the Fourier series in the L−metric. This condition is sufficient under some additional assumptions for Fourier coefficients. In fact, in this paper we generalize some results of A. S. Belov [1]. AMS subject classifications: 42A16, 42A20.