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.
Categories: math.SP