Strong Cesàro |C, 1, 1| summability and statistical convergence of double orthogonal series

Abstract

Let (X,F, μ) be a finite positive measure space and {ϕj,k(x): j, k = 1, 2,...} be a double orthonormal system of real-valued functions on X. We investigate the pointwise convergence of the double orthogonal series (2.1) in Pringsheim’s sense and in the regular sense introduced by Hardy, as well as its Cesaro (C, 1, 1) summability and its strong Cesaro |C, 1, 1| summability. In our main theorem (Theorem 2 in Section 3 below) we extend a previous result of Borgen [2] from single to double orthogonal series. The key ingredient of our proof is the extension of the familiar Kronecker lemma from single to double sequences of numbers (see in [8, Theorem 1]. As an application of our Theorem 2, we are able to conclude the a.e. statistical convergence of the double orthogonal series (2.1) under a weaker condition than (2.5) in the Rademacher–Menshov theorem (see Theorem 3 in the last Section 6).