Nearly optimal Kolmogorov widths under holomorphic mappings
Abstract
This paper establishes essentially optimal asymptotic bounds for Kolmogorov widths under holomorphic mappings between complex Banach spaces. Given a compact parameter set whose Kolmogorov widths decay algebraically with rate s, we prove that the widths of its image under a holomorphic mapping decay algebraically with every rate t<s, thereby answering an open question raised by Cohen and DeVore. As an application, we obtain a sharp characterization of the approximability of solution manifolds associated with inf-sup stable parametrized PDEs. We also construct an explicit example showing that the arbitrarily small loss in the algebraic decay exponent is unavoidable. Finally, we provide similar characterization of asymptotic bounds for Kolmogorov widths in the exponentially decaying regime. Our analysis involves multilinear Taylor expansion in Banach spaces and a novel block dyadic expansion-truncation technique.
Related
- Holomorphic motions, Assouad dimension and quasiconformal mappings
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- A sharp bound for sampling widths in the uniform norm: kernel D-optimal designs and oversampling
BibTeX
@misc{li2026nearly,
title = {{Nearly optimal Kolmogorov widths under holomorphic mappings}},
author = {Yuwen Li and Guozhi Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.05581},
eprint = {2608.05581},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.05581},
}
RIS
TY - UNPB
TI - Nearly optimal Kolmogorov widths under holomorphic mappings
AU - Yuwen Li
AU - Guozhi Zhang
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2608.05581
UR - https://arxiv.org/abs/2608.05581
AB - This paper establishes essentially optimal asymptotic bounds for Kolmogorov widths under holomorphic mappings between complex Banach spaces. Given a compact parameter set whose Kolmogorov widths decay algebraically with rate s, we prove that the widths of its image under a holomorphic mapping decay algebraically with every rate t<s, thereby answering an open question raised by Cohen and DeVore. As an application, we obtain a sharp characterization of the approximability of solution manifolds associated with inf-sup stable parametrized PDEs. We also construct an explicit example showing that the arbitrarily small loss in the algebraic decay exponent is unavoidable. Finally, we provide similar characterization of asymptotic bounds for Kolmogorov widths in the exponentially decaying regime. Our analysis involves multilinear Taylor expansion in Banach spaces and a novel block dyadic expansion-truncation technique.
ER -