Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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

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.

References [Enhancements On Off] (What's this?)

  • 1. N. I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, Hafner Publishing Company, New York, 1965. MR 32:1518
  • 2. S. A. Denisov, On the continuous analog of Rakhmanov's theorem for orthogonal polynomials, J. Funct. Anal. 198 (2003), 465-480.
  • 3. Y. L. Geronimus, Orthogonal Polynomials: Estimates, asymptotic formulas, and series of polynomials orthogonal on the unit circle and on an interval, Consultants Bureau, New York, 1961. MR 24:A3469
  • 4. Ya. L. Geronimus, Polynomials Orthogonal on a Circle and their Applications, Amer. Math. Soc. Transl. (1), Vol. 3, Providence, Rhode Island, 1954, 1-78. MR 15:869i
  • 5. A. Máté, P. Nevai, and V. Totik, Asymptotics for the ratio of leading coefficients of orthogonal polynomials on the unit circle, Constr. Approx. 1 (1985), 63-69. MR 85j:42045
  • 6. P. Nevai, Research problems in orthogonal polynomials, Approximation Theory 6, Vol. 2 (C. K. Chui et al., eds.), pp. 449-489, Academic Press, Boston, 1989. MR 91m:42023
  • 7. P. Nevai, Orthogonal polynomials, Memoirs Amer. Math. Soc., Vol. 18, No. 213, Amer. Math. Soc., Providence, RI, 1979. MR 80k:42025
  • 8. P. Nevai, Weakly convergent sequences of functions and orthogonal polynomials, J. Approx. Theory 65 (1991) No. 3, 322-340. MR 92f:42031
  • 9. E. M. Nikishin, The discrete Sturm-Liouville operator and some problems of function theory, Trudy Sem. Petrovsk. 10 (1984), 3-77 (in Russian), J. Soviet Math. 35 (1986), 2679-2744. MR 86h:39007
  • 10. E. A. Rakhmanov, On the asymptotics of the ratio of orthogonal polynomials. II, Math. USSR Sb. 46 (1983), 105-117.
  • 11. F. Riesz and B. Sz.-Nagy, Functional Analysis, Frederick Ungar Publishing, New York, 1955. MR 17:175i
  • 12. G. Szego, Orthogonal Polynomials, Amer. Math. Soc. Colloquium Publications, Vol. 23, Providence, RI, 1975. MR 51:8724

Similar Articles

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

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