Abstract. We examine the problem of equivalence of the following conditions: $$\sum^\infty_{m=0} \alpha_m\Bigl(\sum^{\nu_{m+1}}_{n=\nu_m+1}|c_n|^q\Bigr)^{p/q}< \infty\hbox{ and } \sum^\infty_{m=1} \beta_m \Bigl(\sum^m_{n=1} \gamma_n|c_n|^q\Bigr)^{p/q} < \infty.$$ Using these results and some known theorems we give new conditions for $|R,\lambda,\gamma(t)|_k$-summability of orthogonal series, furthermore some structural conditions pertaining to Fourier series.

AMS Subject Classification (1991): 40A05, 42A28, 42C15

Received February 22, 1994. (Registered under 5647/2009.)