Sparse systems of functions and quasi-analytic classes
Logic
2023-12-07 v1 Complex Variables
Functional Analysis
Abstract
We provide a new characterization of quasi-analyticity of Denjoy-Carleman classes, related to \emph{Wetzel's Problem}. We also completely resolve which Denjoy-Carleman classes carry \emph{sparse systems}: if the Continuum Hypothesis (CH) holds, \textbf{all} Denjoy-Carleman classes carry sparse systems; but if CH fails, a Denjoy-Carleman class carries a sparse system if and only if it is not quasi-analytic. As corollaries, we extend results of \cite{MR3552748} and \cite{CodyCoxLee} about non-existence of "anonymous predictors" for real functions.
Cite
@article{arxiv.2312.03238,
title = {Sparse systems of functions and quasi-analytic classes},
author = {Marco Aldi and Jeffrey Buffkin and Cody Cline and Sean Cox},
journal= {arXiv preprint arXiv:2312.03238},
year = {2023}
}