Matrix transformations of power series

Abstract

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