Abstract. In this paper, we shall give sufficient conditions for the convergence of the lacunary Fourier series having the following form $$\sum_{k=- \infty }^{\infty }k^\delta\varphi \left(|\hat{f}(n_k)| \right )< \infty, $$ where $\delta $ is a non-negative number and $\varphi $ is an increasing and concave function.

AMS Subject Classification (1991): 42A28, 42A55

Keyword(s): Absolute convergence, Lacunary series

Received September 11, 2002, and in revised form December 30, 2002. (Registered under 2923/2009.)