ACTA issues

On absolute summability of orthogonal series by the generalized method of the summatorial function

O. A. Ziza

Acta Sci. Math. (Szeged) 64:3-4(1998), 559-573

Abstract. We consider the summability of orthogonal series (OS) $\sum c_nf_n(x)$, $f_n\in L^2[0,1]$, $\{c_n\} \in l^2$, by the generalized methods of the summatorial function $(SF^*,\varphi,\lambda )$. We prove a theorem on a condition sufficient for the absolute summability of OS a.e. by these methods. This theorem is an analogue of the theorem on Ces?ro methods in \cite8 and also a generalization of one of the author's theorems in [11]. Applications to the particular methods of the form $(SF^*,\varphi,\lambda )$ are indicated, including the Riesz and the Bernstein--Rogosinski methods.

AMS Subject Classification (1991): 42C15

Received May 23, 1998. (Registered under 3309/2009.)