Monomial convergence for holomorphic functions on ℓr

Abstract

Let $$\mathcal{F}$$ be either the set of all bounded holomorphic functions or the set of all m-homogeneous polynomials on the unit ball of lr. We give a systematic study of the sets of all u ∊ lr for which the monomial expansion $$\sum\nolimits_\alpha {{{{\partial ^\alpha }f(0)} \over {\alpha !}}{u^\alpha }} $$ of every f ∊ $$\mathcal{F}$$ converges. Inspired by recent results from the general theory of Dirichlet series, we establish as our main tool independently interesting, upper estimates for the unconditional basis constants of spaces of polynomials on lr spanned by finite sets of monomials.