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Licensed Unlicensed Requires Authentication Published by De Gruyter July 6, 2009

Domains of convergence for monomial expansions of holomorphic functions in infinitely many variables

Andreas Defant, Manuel Maestre and Christopher Prengel
From the journal

Abstract

Let ℱ(R) be a set of holomorphic functions on a Reinhardt domain R in a Banach sequence space (as e.g. all holomorphic functions or all m-homogeneous polynomials on the open unit ball of ). We give a systematic study of the sets dom ℱ(R) of all zR for which the monomial expansion of every ƒ ∈ ℱ(R) converges. Our results are based on and improve the former work of Bohr, Dineen, Lempert, Matos and Ryan. In particular, we show that up to any ε > 0 is the unique Banach sequence space X for which the monomial expansion of each holomorphic function ƒ converges at each point of a given Reinhardt domain in X. Our study shows clearly why Hilbert's point of view to develop a theory of infinite dimensional complex analysis based on the concept of monomial expansion, had to be abandoned early in the development of the theory.

Received: 2008-04-02
Published Online: 2009-07-06
Published in Print: 2009-September

© Walter de Gruyter Berlin · New York 2009