Year: 2026
Type: preprint
Abstract: A classical phenomenon distinguishing infinite-dimensional holomorphy from its finite-dimensional counterpart is that monomial expansions of holomorphic functions need not converge absolutely. In this paper, we systematically investigate monomial convergence on complex Banach sequence spaces. We establish general structural results and give a negative answer to Lempert's problem. We then determine the domains of absolute monomial convergence for holomorphic functions on the classical Banach sequence spaces and obtain a complete classification for continuous homogeneous polynomials. As a further application, we establish a sharp comparison between two notions of convergence radius for homogeneous Taylor expansions on $\ell^1$. The methods developed here are of independent interest and may be applied in other places.
Source: arXiv (Cornell University)
Author: Wang, Zhouzhe
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220892061