On L-convergence of trigonometric series

In the present paper we consider the trigonometric series with ( β , r ) -general monotone and ( β , r ) -rest bounded variation coefficients. Necessary and sufficient conditions of L-convergence for such series are obtained in terms of the coefficients. Moreover, we generalize and extend the Tikhonov results [J. Math. Anal. Appl. 347 (2008) 416–427] to the class GM ( β , r ) or the class RBVS ( β , r ) .