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Sum rules and the Szegő condition for orthogonal polynomials on the real line. (English) Zbl 1046.42017
This densely written, unifying paper contains a wealth of results on the relations between the following three (equivalent) notions: 1. non-trivial probability measures \(\nu\) of bounded support on \(\mathbb{R}\) with infinitely many points, 2. orthogonal polynomials associated to geometrically bounded moments, 3. bounded Jacobi-matrices.
Introducing the Jacobi-matric \(J\) by \[ J=\begin{pmatrix} b_1 & a_1 & 0 & \dots \cr a_1 & b_2 & a_2 & \dots \cr 0 & a_2 & b_3 & \dots \cr \dots & \dots & \dots & \ldots\end{pmatrix} \;(a_n>0), \] \(\nu\) will be the spectral measure of \(\delta_1\in \mathbb{L}^2(\mathbb{Z^{+}})\) and the \(P_n(x)\) denote the orthonormal polynomials. The Szegő integral will be given in the form \[ Z(J)={1\over 2\pi}\int_{-2}^2\,\log \left({\sqrt{4-E^2}\over 2\pi \text{d}\nu /dE}\right){dE\over \sqrt{4-E^2}} \] (this is \(c-I_S\) where \(I_S\) is the Szegő integral with \(d\nu_{\text{ac}}/dE\) as argument of \(\log\)).
One of the main results (used to prove general results from the existing literature and to derive results concerning the failure of the Szegő condition at the points \(\pm 2\)) is given by the following. Let \[ A_0(J)=\lim_{N\to\infty}\, \left( -\sum_{n=1}^N\,\log{(a_n)}\right) \] exist (\(\pm \infty\) allowed), let \[ {\mathcal E}(J)=\sum_{\pm}\,\sum_j \log{\left[\tfrac12(| E_j^{\pm}| + \sqrt{(E_j^{\pm})^2-4}) \right]}, \] \(\sigma_{\text{ess}}(J)=[-2,2]\), \(J\) has only eigenvalues \(E_j^{\pm}\) outside the interval \([-2,2]\) with multiplicity one and ordered by \(E_j^{+}\) decreasing in \(j\), all greater than \(2\) and \(E_j^{-}\) increasing in \(j\), all smaller than \(-2\). Then: if two of the three quantities \(A_0(J), {\mathcal E}(J)\) and \(Z(J)\) are finite, then all three are and, moreover, we have \[ A_0(J)+ {\mathcal E}(J)=Z(J). \]
A few of the subsequent results are, for instance: A. If \(\lim_{n\to\infty}n(a_n-1)=\alpha,\;\lim_{n\to\infty}nb_n =\beta\) exist and are finite with \(| \beta| >2\alpha\), then \(Z(J)=\infty\). B. If \(b_n\geq 0\) and \(\sum_{n=1}^{\infty}| a_n-1| <\infty\), then \({\mathcal E}(J)<\infty \Leftrightarrow \sum_{n=1}^{\infty}b_n<\infty\). C. \(\sum_{n=1}^{\infty} [(a_n-1)^2+b_n^2]<\infty, \limsup_{N\to\infty}\left(\sum_{n=1}^N (a_n-1\pm b_n/2)\right)\) \(=\infty\), then the Szegő condition fails at \(\pm 2\).

MSC:
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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