Eigenvalue asymptotics for a class of multi-variable Hankel matrices

Abstract

Abstract A one-variable Hankel matrix H a {H}_{a} is an infinite matrix H a = [ a ( i + j ) ] i , j ≥ 0 {H}_{a}={\left[a\left(i+j)]}_{i,j\ge 0} . Similarly, for any d ≥ 2 d\ge 2 , a d d -variable Hankel matrix is defined as H a = [ a ( i + j ) ] {H}_{{\bf{a}}}=\left[{\bf{a}}\left({\bf{i}}+{\bf{j}})] , where i = ( i 1 , … , i d ) {\bf{i}}=\left({i}_{1},\ldots ,{i}_{d}) and j = ( j 1 , … , j d ) {\bf{j}}=\left({j}_{1},\ldots ,{j}_{d}) , with i 1 , … , i d , j 1 , … , j d ≥ 0 {i}_{1},\ldots ,{i}_{d},{j}_{1},\ldots ,{j}_{d}\ge 0 . For γ > 0 \gamma \gt 0 , Pushnitski and Yafaev proved that the eigenvalues of the compact one-variable Hankel matrices H a {H}_{a} with a ( j ) = j − 1 ( log j ) − γ a\left(j)={j}^{-1}{\left(\log j)}^{-\gamma } , for j ≥ 2 j\ge 2 , obey the asymptotics λ n ( H a ) ∼ C γ n − γ {\lambda }_{n}\left({H}_{a})\hspace{0.33em} \sim \hspace{0.33em}{C}_{\gamma }{n}^{-\gamma } , as n → + ∞ n\to +\infty , where the constant C γ {C}_{\gamma } is calculated explicitly. This article presents the following d d -variable analogue. Let γ > 0 \gamma \gt 0 and a ( j ) = j − d ( log j ) − γ a\left(j)={j}^{-d}{\left(\log j)}^{-\gamma } , for j ≥ 2 j\ge 2 . If a ( j 1 , … , j d ) = a ( j 1 + ⋯ + j d ) {\bf{a}}\left({j}_{1},\ldots ,{j}_{d})=a\left({j}_{1}+\cdots +{j}_{d}) , then H a {H}_{{\bf{a}}} is compact and its eigenvalues follow the asymptotics λ n ( H a ) ∼ C d , γ n − γ {\lambda }_{n}\left({H}_{{\bf{a}}})\hspace{0.33em} \sim \hspace{0.33em}{C}_{d,\gamma }{n}^{-\gamma } , as n → + ∞ n\to +\infty , where the constant C d , γ {C}_{d,\gamma } is calculated explicitly.