An analog of Hahn's theorem for infinite products (Q915034)

Let \(A=(a_{nk})\) be an infinite matrix. An infinite product (1) \(\prod^{\infty}_{k=1}(1+u_ k)\) is said to be A-summable to the number a provided that \(\lim_{n\to \infty}A_ n=a\), where \(A_ n=\prod^{\infty}_{k=1}(1+a_{nk}u_ k)\) \((n=1,2,...)\). The author gives necessary and sufficient conditions for A-summability of the product (1) under the assumption that \(\sum^{\infty}_{k=2}| u_ k-u_{k-1}| <+\infty\).