Power series with H.-O. gaps; Tauberian theorems (Q897416)

Let \(A=[\alpha _{nk}]\) be a triangular matrix, and let \(\{s_n(z)\}\) be the sequence of partial sums of a power series with radius of convergence 1. The series is said to be \(A\)-summable, if the \(A\) transform \(\{t_n(z)\}\) of \(\{s_n(z)\}\) is convergent. The authors prove two results on the convergence of subsequences of \(\{s_n(z)\}\), under the assumption of certain \(A\)-summability properties of \(\{s_n(z)\}\) outside the unit disk, and a condition on the elements of \(A\); cf. [\textit{A. Ostrowski}, Berl. Ber. 1923, 185--192 (1923; JFM 49.0230.03)].