A lower bound in Nehari's theorem on the polydisc
Complex Variables
2012-11-13 v1 Classical Analysis and ODEs
Functional Analysis
Abstract
By theorems of Ferguson and Lacey (d=2) and Lacey and Terwilleger (d>2), Nehari's theorem is known to hold on the polydisc D^d for d>1, i.e., if H_\psi is a bounded Hankel form on H^2(D^d) with analytic symbol \psi, then there is a function \phi in L^\infty(\T^d) such that \psi is the Riesz projection of \phi. A method proposed in Helson's last paper is used to show that the constant C_d in the estimate \|\phi\|_\infty\le C_d \|H_\psi\| grows at least exponentially with d; it follows that there is no analogue of Nehari's theorem on the infinite-dimensional polydisc.
Cite
@article{arxiv.1107.0175,
title = {A lower bound in Nehari's theorem on the polydisc},
author = {Joaquim Ortega-Cerdá and Kristian Seip},
journal= {arXiv preprint arXiv:1107.0175},
year = {2012}
}