Abstract. We consider summability of orthogonal series (OS) $\sum c_nf_n(x)$, $x\in[0,1]$, a.e. by methods of class $\{ (\varphi,\lambda )\}$, which includes many concrete methods of summability. Assuming that $\sum c_n^2l^2(n)< \infty,$ $0< l(n)\uparrow\infty $, we obtain estimates of the rate of convergence to $0$ of the deviation $|f(x)-\sigma(x,u)|$ in terms of the rate of increase of $l(n)$; here $f$ is the sum of the OS in $L^2[0,1]$, and $\sigma(x,u)$ are the $(\varphi,\lambda )$--means of the OS.

AMS Subject Classification (1991): 42C15

Received May 10, 1996 and in revised form September 30, 1996. (Registered under 6107/2009.)