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

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 z ∈ R 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.