On Rakhmanov's theorem for Jacobi matrices

Author:
Sergey A. Denisov

Journal:
Proc. Amer. Math. Soc. **132** (2004), 847-852

MSC (2000):
Primary 47B36

Published electronically:
July 7, 2003

MathSciNet review:
2019964

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove Rakhmanov's theorem for Jacobi matrices without the additional assumption that the number of bound states is finite. This result solves one of Nevai's open problems.

**1.**N. I. Akhiezer,*The classical moment problem and some related questions in analysis*, Translated by N. Kemmer, Hafner Publishing Co., New York, 1965. MR**0184042****2.**S. A. Denisov, On the continuous analog of Rakhmanov's theorem for orthogonal polynomials,*J. Funct. Anal.***198**(2003), 465-480.**3.**L. Ya. Geronimus,*Orthogonal polynomials: Estimates, asymptotic formulas, and series of polynomials orthogonal on the unit circle and on an interval*, Authorized translation from the Russian, Consultants Bureau, New York, 1961. MR**0133643****4.**Ya. L. Geronimus,*Polynomials orthogonal on a circle and their applications*, Amer. Math. Soc. Translation**1954**(1954), no. 104, 79. MR**0061706****5.**Attila Máté, Paul Nevai, and Vilmos Totik,*Asymptotics for the ratio of leading coefficients of orthonormal polynomials on the unit circle*, Constr. Approx.**1**(1985), no. 1, 63–69. MR**766095**, 10.1007/BF01890022**6.**Paul Nevai,*Research problems in orthogonal polynomials*, Approximation theory VI, Vol. II (College Station, TX, 1989) Academic Press, Boston, MA, 1989, pp. 449–489. MR**1091045****7.**Paul G. Nevai,*Orthogonal polynomials*, Mem. Amer. Math. Soc.**18**(1979), no. 213, v+185. MR**519926**, 10.1090/memo/0213**8.**Paul Nevai,*Weakly convergent sequences of functions and orthogonal polynomials*, J. Approx. Theory**65**(1991), no. 3, 322–340. MR**1109411**, 10.1016/0021-9045(91)90095-R**9.**E. M. Nikishin,*The discrete Sturm-Liouville operator and some problems of function theory*, Trudy Sem. Petrovsk.**10**(1984), 3–77, 237 (Russian, with English summary). MR**778879****10.**E. A. Rakhmanov, On the asymptotics of the ratio of orthogonal polynomials. II,*Math. USSR Sb.***46**(1983), 105-117.**11.**Frigyes Riesz and Béla Sz.-Nagy,*Functional analysis*, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron. MR**0071727****12.**Gábor Szegő,*Orthogonal polynomials*, 4th ed., American Mathematical Society, Providence, R.I., 1975. American Mathematical Society, Colloquium Publications, Vol. XXIII. MR**0372517**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
47B36

Retrieve articles in all journals with MSC (2000): 47B36

Additional Information

**Sergey A. Denisov**

Affiliation:
Department of Mathematics, California Institute of Technology, 253-37, Pasadena, California 91125

Email:
denissov@caltech.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-03-07157-0

Received by editor(s):
October 16, 2002

Received by editor(s) in revised form:
November 8, 2002

Published electronically:
July 7, 2003

Communicated by:
Andreas Seeger

Article copyright:
© Copyright 2003
American Mathematical Society