Matrix transformations of power series

Abstract:

We consider the sequence of transforms $({g_n})$ of a power series $\sum \nolimits _{n = 0}^\infty {{a_n}{z^n}}$ given by ${g_n}(z): = \sum \nolimits _{k = 0}^\infty {{b_{nk}}{a_k}{z^k}}$. We establish necessary and sufficient conditions on the matrix $({b_{nk}})$ for the sequence $({g_n})$ to converge uniformly on compact subsets of the disk ${D_P}: = \{ z:|z| < P\}$ to a function holomorphic on ${D_P}$.