Some limit relations for multidimensional positive-definite Toeplitz matrices (Q1111151)

Let \(G_ 1\subset G_ 2\subset..\). be an increasing sequence of bounded regions in \({\mathbb{R}}^ p_+=\{\bar x=(x_ 1,...,x_ n):\) \(\forall n\) \(x_ n>0\}\) and \(G_ 1\cup...\cup G_ n\cup...={\mathbb{R}}^ p_+\), \(G'_ n=G\cap {\mathbb{Z}}^ p\). Let \(I_ n=\| g^{(n)}_{k,\ell}\|\) \(((k,\ell)\in G'_ n)\) be a positive Toeplitz matrix. \(I_ n^{-1}=\| \tilde g^{(n)}_{k,\ell}\|\) and \(d_ n=\sum_{(k,\ell)\in G'_ n}\tilde g^{(n)}_{k,\ell}.\) The aim of this paper is to study of asymptotic behaviour of \(d_ n\) when \(n\to \infty\).