Power series with H.-O. Gaps; Tauberian theorems

Abstract

Let Σ ν = 0 ∞ a ν z ν be a power series with radius of convergence 1, and let s n (z) = Σ ν = 0 n a ν z ν denote its partial sums. For a given triangular matrix A = [α nν ] we consider the A-transforms σ n (z) = Σ ν = 0 n α nν s ν (z), and prove two Tauberian theorems of the following type: from certain summability properties of {σ n (z)} outside the unit disk and a condition on the entries α nν the convergence of a subsequence \(\left\{ {s_{n_k } \left( z \right)} \right\}\) is concluded.