zbMATH — the first resource for mathematics

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 \(\pm 2\).

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.)
Full Text: DOI arXiv
[1] Askey, R., Ismail, M.: Recurrence relations, continued fractions, and orthogonal polynomials. Mem. Am. Math. Soc. 49, (1984) · Zbl 0548.33001
[2] Case, K.M.: Orthogonal polynomials. II. J. Math. Phys. 16, 1435–1440 (1975) · Zbl 0304.42015
[3] Charris, J., Ismail, M.E.H.: On sieved orthogonal polynomials, V. Sieved Pollaczek polynomials. SIAM J. Math. Anal. 18, 1177–1218 (1987) · Zbl 0645.33018
[4] Damanik, D., Hundertmark, D., Simon, B.: Bound states and the Szego condition for Jacobi matrices and Schrödinger operators. J. Funct. Anal., to appear · Zbl 1049.47032
[5] Deift, P., Killip, R.: On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials. Commun. Math. Phys. 203, 341–347 (1999) · Zbl 0934.34075
[6] Denisov, S.A.: On the coexistence of absolutely continuous and singular continuous components of the spectral measure for some Sturm-Liouville operators with square summable potentials. J. Diff. Eqs. 191, 90–104 (2003) · Zbl 1032.34079
[7] Dombrowski, J., Nevai, P.: Orthogonal polynomials, measures and recurrence relations. SIAM J. Math. Anal. 17, 752–759 (1986) · Zbl 0595.42011
[8] Figotin, A., Pastur, L.: Spectra of random and almost-periodic operators. Berlin: Springer-Verlag, 1992 · Zbl 0752.47002
[9] Gončar, A.A.: On convergence of Padé approximants for some classes of meromorphic functions. Math. USSR Sb. 26, 555–575 (1975) · Zbl 0341.30029
[10] Hundertmark, D., Simon, B.: Lieb-Thirring inequalities for Jacobi matrices. J. Approx. Theory 118, 106–130 (2002) · Zbl 1019.39013
[11] Killip, R., Simon, B.: Sum rules for Jacobi matrices and their applications to spectral theory. Ann. Math. 158, 253–321 (2003) · Zbl 1050.47025
[12] Laptev, A., Naboko, S., Safronov, O.: On new relations between spectral properties of Jacobi matrices and their coefficients. Commun. Math. Phys., to appear · Zbl 1135.47303
[13] Nevai, P.: Orthogonal polynomials defined by a recurrence relation. Trans. Am. Math. Soc. 250, 369–384 (1979) · Zbl 0413.42015
[14] Nevai, P.: Orthogonal polynomials. Mem. Am. Math. Soc. 18(213), 185 pp (1979) · Zbl 0405.33009
[15] Nevai, P.: Géza Freud, orthogonal polynomials and Christoffel functions. A case study. J. Approx. Theory 48, 3–167 (1986) · Zbl 0606.42020
[16] Nikishin, E.M.: Discrete Sturm-Liouville operators and some problems of function theory. J. Sov. Math. 35, 2679–2744 (1986) · Zbl 0605.34023
[17] Peherstorfer, F., Yuditskii, P.: Asymptotics of orthonormal polynomials in the presence of a denumerable set of mass points. Proc. Am. Math. Soc. 129, 3213–3220 (2001) · Zbl 0976.42012
[18] Pollaczek, F.: Sur une généralisation des polynomes de Legendre. C. R. Acad. Sci. Paris 228, 1363–1365 (1949) · Zbl 0041.03502
[19] Pollaczek, F.: Systèmes de polynomes biorthogonaux qui généralisent les polynomes ultrasphériques. C. R. Acad. Sci. Paris 228, 1998–2000 (1949)
[20] Pollaczek, F.: Sur une famille de polynômes orthogonaux qui contient les polynômes d’Hermite et de Laguerre comme cas limites. C. R. Acad. Sci. Paris 230, 1563–1565 (1950) · Zbl 0039.07704
[21] Rudin, W.: Real and Complex Analysis, 3rd edn. New York: Mc-Graw Hill, 1987 · Zbl 0925.00005
[22] Shohat, J.A.: Théorie Générale des Polinomes Orthogonaux de Tchebichef. Mémorial des Sciences Mathématiques, Vol. 66. Paris: 1934, pp. 1–69
[23] Simon, B.: The classical moment problem as a self-adjoint finite difference operator. Adv. Math. 137, 82–203 (1998) · Zbl 0910.44004
[24] Szego, G.: Beiträge zue Theorie der Toeplitzschen Formen, I, II. Math. Z. 6, 167–202 (1920); 9, 167–190 (1921) · JFM 47.0391.04
[25] Szego, G.: On certain special sets of orthogonal polynomials. Proc. Am. Math. Soc. 1, 731–737 (1950) · Zbl 0041.39202
[26] Szego, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society, Colloquium Publications, Vol. XXIII. Providence, RI: American Mathematical Society, 1975
[27] Verblunsky, S.: On positive harmonic functions. Proc. London Math. Soc. 40, 290–320 (1935) · Zbl 0013.15702
[28] Zlatoš, A.: The Szego condition for Coulomb Jacobi matrices. J. Approx. Theory 121, 119–142 (2003) · Zbl 1020.15026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.