On Weighted Lp Integrability of Functions Defined by Trigonometric Series

We introduce a new class of sequences called GM¯θr and give a sufficient and necessary condition for weighted Lp integrability of trigonometric series with coefficients to belong to the above class. This is a generalization of the result proved by M. Dyachenko and S. Tikhonov (2009). Then we discuss the relations among the weighted best approximation and the coefficients of trigonometric series. Moreover, we extend the results of B. Wei and D. Yu (2009) to the class GM¯θr