The Banach property of matrices (Q1187333)

Let \(B=(b_{n,k})\) be a matrix for which \(\sum b^ 2_{n,k}\to0\) as \(n\to\infty\). Further, let \(\varphi=\{\varphi_ k(x)\}\) be any orthonormal system on \((0,1)\). \(B\) is said to have the Banach property if for any \(\varphi\), \(b_ n(\varphi;x):=\sum_ k b_{n,k}\varphi_ k(x)\to 0\) a.e. in \((0,1)\). The author gives sufficient conditions on \(B\) to have the Banach property. Examples illustrate the conditions provided.