Monomial convergence for holomorphic functions on $\ell\_r$

TL;DR

This paper systematically studies the convergence of monomial expansions for holomorphic functions and polynomials on the unit ball of l_r, providing new estimates for basis constants inspired by Dirichlet series theory.

Contribution

It introduces a systematic analysis of monomial convergence sets and establishes new upper bounds for basis constants in polynomial spaces on l_r.

Findings

Identifies sets of points where monomial expansions converge for all functions in 

Provides upper estimates for unconditional basis constants of polynomial spaces

Connects results to recent developments in Dirichlet series theory

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 $\ell\_r$. We give a systematic study of the sets of all $u\in\ell\_r$ for which the monomial expansion $\sum\_{\alpha}\frac{\partial^\alpha f(0)}{\alpha !}u^\alpha$ of every $f\in\mathcal F$ 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 $\ell\_r$ spanned by finite sets of monomials.