A Helson matrix with explicit eigenvalue asymptotics

Abstract

A Helson matrix (also known as a multiplicative Hankel matrix) is an infinite matrix with entries $\{a(jk)\}$ for $j,k\geq1$. Here the $(j,k)$'th term depends on the product $jk$. We study a self-adjoint Helson matrix for a particular sequence $a(j)=(\sqrt{j}\log j(\log\log j)^\alpha))^{-1}$, $j\geq 3$, where $\alpha>0$, and prove that it is compact and that its eigenvalues obey the asymptotics $\lambda_n\sim\varkappa(\alpha)/n^\alpha$ as $n\to\infty$, with an explicit constant $\varkappa(\alpha)$. We also establish some intermediate results (of an independent interest) which give a connection between the spectral properties of a Helson matrix and those of its continuous analogue, which we call the integral Helson operator.