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PreprintCC BY 4.0arXiv

Domains of Monomial Convergence in Banach Sequence Spaces

  • Zhouzhe Wang
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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 ℓ1. The methods developed here are of independent interest and may be applied in other places.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026domains,
  title = {{Domains of Monomial Convergence in Banach Sequence Spaces}},
  author = {Zhouzhe Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.08873},
  eprint = {2610.08873},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.08873},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Domains of Monomial Convergence in Banach Sequence Spaces
AU  - Zhouzhe Wang
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2610.08873
UR  - https://arxiv.org/abs/2610.08873
AB  - 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 ℓ^1. The methods developed here are of independent interest and may be applied in other places.
ER  -