Eigenvalues of Toeplitz matrices associated with orthogonal polynomials
In this significant paper, the author extends earlier results of Grenander, Szegö and Nevai relating zero distribution of orthogonal polynomials to the distribution of eigenvalues of related Toeplitz matrices. His results treat simultaneously orthogonal polynomials on finite and infinite intervals. Let \[ p_ n(x)=\gamma_ n\prod^{n}_{j=1}(x-x_{jn}),\quad \gamma_ n>0, \] denote the nth orthonormal polynomial for a positive measure \(d\alpha\) (x) on \({\mathbb{R}}\), \(n=0,1,2,..\).. Let g be a real valued function such that for \(k=0,1,2,...,x^ kg(x)\in L_ 1(d\alpha)\). Let \(\{c_ n\}^{\infty}_{n=1}\) be a sequence of positive numbers and introduce the \(n\times n\) Toeplitz matrix \[ T_ n(g)=[\int^{\infty}_{- \infty}p_ j(x)p_ k(x)g(x/c_ n)d\alpha (x)]^{n-1}_{j,k=0}, \] with eigenvalues \(\{\Lambda_{jn}\}^ n_{j=1}\), repeated according to multiplicity. Define probability measures \(\mu_ n\) and \(\nu_ n\) placing mass 1/n at \(x_{ij}/c_ n\) and \(\Lambda_{jn}\) respectively: So \[ \mu_ n=n^{-1}\sum^{n}_{j=1}\delta_{x_{jn}/c_ n};\quad \nu_ n=n^{-1}\sum^{n}_{j=1}\delta_{\Lambda_{jn}}, \] (\(\delta=\) Dirac delta). The main theorem states that if \(\mu_ n\) converges weakly to \(\mu\), and \(c_ n^{-1}(\gamma_{n-1}/\gamma_ n)\) is \(o(n^{1/2})\), and g is bounded with discontinuities in a set of \(\mu\)- measure zero, then \(\nu_ n\) converges weakly to a probability measure \(\nu\). More precisely, for every f continuous on \({\mathbb{R}}\), \[ \lim_{n\to \infty}n^{- 1}\sum^{n}_{j=1}f(\Lambda_{jn})=\int^{\infty}_{- \infty}f(g(x))d\mu (x). \]
scientific article; zbMATH DE number 4032104
scientific article