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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.)
##### Keywords:
sum rules; Szegő condition; orthogonal polynomials
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