Convergence on segments and convexity theorems (Q916016)

Let \(A=(a_{mn})\) be a regular normal positive matrix. A sequence \(\{S_ n\}\) of complex numbers is said to be A-bounded (A-convergent) provided that \(\sup_{0\leq m<\infty}| t_ m| <+\infty\) \((\lim_{n\to \infty}t_ m=L\in C)\), where \(t_ m=\sum^{m}_{n=0}a_{mn}S_ n\) \((m=0,1,...)\). Denote by \({\mathcal A}_ b\) and \({\mathcal A}_ c\) the set of all A-bounded and A-convergent sequences, respectively. Using these sets the author defines classes \(\Delta_ s\) and \(\delta_ s\) of matrices. In the paper the convexity theorem well-known from (C,k)-summability \((k>-1)\) is extended to matrix methods corresponding to matrices of \(\Delta_ s\) and \(\delta_ s\).