Eigenvalue asymptotics for a class of multi-variable Hankel matrices

Abstract

A one-variable Hankel matrix $H_a$ is an infinite matrix $H_a=[a(i+j)]_{i,j\geq0}$. Similarly, for any $d\geq2$, a $d$-variable Hankel matrix is defined as $H_{\mathbf{a}}=[\mathbf{a}(\mathbf{i}+\mathbf{j})]$, where $\mathbf{i}=(i_1,\dots,i_d)$ and $\mathbf{j}=(j_1,\dots,j_d)$, with $i_1,\dots,i_d,j_1,\dots,j_d\geq0$. For $\gamma>0$, A. Pushnitski and D. Yafaev proved that the eigenvalues of the compact one-variable Hankel matrices $H_a$ with $a(j)=j^{-1}(\log j)^{-\gamma}$, for $j\geq2$, obey the asymptotics $\lambda_n(H_a)\sim C_\gamma n^{-\gamma}$, as $n\to+\infty$, where the constant $C_\gamma$ is calculated explicitly. This paper presents the following $d$-variable analogue. Let $\gamma>0$ and $a(j)=j^{-d}(\log j)^{-\gamma}$, for $j\geq2$. If $\mathbf{a}(j_1,\dots,j_d)=a(j_1+\dots+j_d)$, then $H_{\mathbf{a}}$ is compact and its eigenvalues follow the asymptotics $\lambda_n(H_{\mathbf{a}})\sim C_{d,\gamma}n^{-\gamma}$, as $n\to+\infty$, where the constant $C_{d,\gamma}$ is calculated explicitly.