Eigenvalue asymptotics for a class of multi-variable Hankel matrices

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