Series expansions of analytic functions. II

James Donnell Buckholtz and Ken Shaw

Vol. 56 (1975), No. 2, 373–384

DOI: 10.2140/pjm.1975.56.373

Abstract

This paper is concerned with series expansions of the form f(z) = ∑ ₀^∞h_kp_k(z), where the functions {p_k} are analytic and satisfy a certain asymptotic condition. Relationships between the space ℱ of expandable functions, the coefficient space ℋ, and the matrix operator B_jk = p_k^(j)(0) are studied, and ℱ is shown to be a Banach space isomorphic to c₀, the space of complex sequences with limit 0. Necessary and sufficient conditions for convergence of ∑ ₀^∞h_kp_k(z) are given in terms of the coefficient sequence h.

Mathematical Subject Classification

Primary: 30A18

Milestones

Received: 11 December 1973

Published: 1 February 1975