On absolutely regular summability methods (Q1105794)

The authors consider the generalisation of a summability matrix to an infinite sequence of summability matrices. If \(A^ i=(a^ i_{n,k})\) is an infinite matrix of complex numbers. Let \(\{A\}\) denote the sequence of matrices \(\{A^ i\}\). For a sequence of complex numbers \(x=\{x_ n\}\), the sequence \(y=\{y^ i_ n\}\) defined by \(y^ i_ n=\sum^{\infty}_{k=0}a^ i_{n,k}x_ k\) is called the \(\{A\}\) transform of x whenever the series converges for all n and i. A sequence x is said to be \(\{A\}\)-summable to L if \(\{y^ i_ n\}\) converges to L as n tends to \(\infty\) uniformly in i. If \(y^ i_{-1}\) is taken to be zero for all i, then the authors say that the family of sequences \(\{y^ i_ n\}\in (\nu)\) if the series \(\sum^{\infty}_{k=0}| y^ i_ n-y^ i_{n-1}|\) converges uniformly in i and \(\lim_{n\to \infty}y^ i_ n\) assumes the same value for all i. If \(x\in (\nu)\) implies that \(\{x^ i_ n\}\in \nu\) then the authors say that \(\{A\}\in (\nu,\nu)\). The authors establish necessary and sufficient conditions on \(\{A\}\) for \(\{A\}\in (\nu,\nu)\).