Skip to main content

Advertisement

SpringerLink
Log in

CMV Matrices and Little and Big −1 Jacobi Polynomials

  • Published:
Constructive Approximation Aims and scope

Abstract

We introduce a new map from polynomials orthogonal on the unit circle to polynomials orthogonal on the real axis. This map is closely related to the theory of CMV matrices. It contains an arbitrary parameter λ which leads to a linear operator pencil. We show that the little and big −1 Jacobi polynomials are naturally obtained under this map from the Jacobi polynomials on the unit circle.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anshelevich, M.: Two-state free Brownian motions. J. Funct. Anal. 260(2), 541–565 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Atia, M., Alaya, J., Ronveaux, A.: Some generalized Jacobi polynomials. Comput. Math. Appl. 45, 843–850 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Atkinson, F.V.: Multiparameter Eigenvalue Problems. Academic Press, New York (1972)

    MATH  Google Scholar 

  4. Badkov, V.M.: Uniform asymptotic representations of orthogonal polynomials. Tr. Mat. Inst. Steklova 164, 6–36 (1983). English translation in Proc. Steklov Inst. Math. 164, 5–41 (1985)

    MathSciNet  MATH  Google Scholar 

  5. Badkov, V.M.: Systems of orthogonal polynomials explicitly represented by the Jacobi polynomials. Math. Notes 42, 858–863 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  6. Beckermann, B., Derevyagin, M., Zhedanov, A.: The linear pencil approach to rational interpolation. J. Approx. Theory 162, 1322–1346 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Belmehdi, S.: Generalized Gegenbauer orthogonal polynomials. J. Comput. Appl. Math. 133, 195–205 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Berti, A., Sri Ranga, A.: Companion orthogonal polynomials: some applications. Appl. Numer. Math. 39, 127–149 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bryc, W., Matysiak, W., Wesolowski, J.: Quadratic harnesses, q-commutations, and orthogonal martingale polynomials. Trans. Am. Math. Soc. 359(11), 5449–5483 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bueno, M.I., Marcellán, F.: Darboux transformation and perturbation of linear functionals. Linear Algebra Appl. 384, 215–242 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cantero, M.J., Moral, L., Velázquez, L.: Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle. Linear Algebra Appl. 362, 29–56 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chihara, T.S.: On co-recursive orthogonal polynomials. Proc. Am. Math. Soc. 8, 899–905 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chihara, T.S.: On kernel polynomials and related systems. Boll. Unione Mat. Ital. 3, 451–459 (1964)

    MathSciNet  Google Scholar 

  14. Chihara, T.S.: An Introduction to Orthogonal Polynomials. Mathematics and Its Applications, vol. 13. Gordon and Breach, New York (1978)

    MATH  Google Scholar 

  15. Chihara, L.M., Chihara, T.S.: A class of nonsymmetric orthogonal polynomials. J. Math. Anal. Appl. 126, 275–291 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  16. Corteel, S., Stanley, R., Stanton, D., Williams, L.: Formulae for Askey-Wilson moments and enumeration of staircase tableaux. Trans. Am. Math. Soc. (to appear). arXiv:1007.5174

  17. Delsarte, P., Genin, Y.: The split Levinson algorithm. IEEE Trans. Acoust. Speech Signal Process. 34, 470–478 (1986)

    Article  MathSciNet  Google Scholar 

  18. Delsarte, P., Genin, Y.: Tridiagonal approach to the algebraic environment of Toeplitz matrices I. Basic results. SIAM J. Matrix Anal. Appl. 12, 220–238 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  19. Delsarte, P., Genin, Y.: Tridiagonal approach to the algebraic environment of Toeplitz matrices. II. Zero and eigenvalue problems. SIAM J. Matrix Anal. Appl. 12(3), 432–448 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  20. Derevyagin, M.S.: The Jacobi matrices approach to Nevanlinna–Pick problems, J. Approx. Theory. J. Approx. Theory 163(2), 117–142 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Derevyagin, M.S., Zhedanov, A.S.: An operator approach to multipoint padé approximations. J. Approx. Theory 157, 70–88 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Essler, F., Rittenberg, V.: Representations of the quadratic algebra and partially asymmetric diffusion with open boundaries. J. Phys. A 29, 3375–3407 (1996). arXiv:cond-mat/9506131

    Article  MathSciNet  MATH  Google Scholar 

  23. Geronimus, J.L.: On some difference equations and associated systems of orthogonal polynomials. Zap. Math. Otd. Phys.-Math. Fac. Kharkov Univ. Kharkov Math. Obchestva XXV, 87–100 (1957) (in Russian)

    Google Scholar 

  24. Gladwell, G.: Inverse Problems in Vibration, 2nd edn. Solid Mechanics and Its Applications, vol. 119. Kluwer Academic, Dordrecht (2004)

    Google Scholar 

  25. Golinskii, B.L.: The V.A. Steklov problem in the theory of orthogonal polynomials. Math. Notes 15, 13–19 (1974)

    Article  MATH  Google Scholar 

  26. Hendriksen, E., van Rossum, H.: Orthogonal Laurent polynomials. Indag. Math., Ser. A 48, 17–36 (1986)

    MATH  Google Scholar 

  27. Kerov, S.: Interlacing measures. In: Kirillov’s Seminar on Representation Theory. Am. Math. Soc. Transl. Ser. 2, vol. 181, p. 3583. Am. Math. Soc., Providence (1998) (English summary)

    Google Scholar 

  28. Killip, R., Nenciu, I.: Matrix models for circular ensembles. Int. Math. Res. Not. 50, 2665–2701 (2004)

    Article  MathSciNet  Google Scholar 

  29. Mason, J.C., Handscomb, D.C.: Chebyshev Polynomials. CRC Press, Boca Raton (2003)

    MATH  Google Scholar 

  30. Schoutens, W., Teugels, J.L.: Lévy processes, polynomials and martingales. Special issue in honor of Marcel F. Neuts. Commun. Stat. Stoch. Models 14(1–2), 335–349 (1998)

    MathSciNet  MATH  Google Scholar 

  31. Simon, B.: Orthogonal Polynomials on the Unit Circle. AMS, Providence (2005)

    Google Scholar 

  32. Sleeman, B.D.: Multiparameter Spectral Theory in Hilbert Space. Pitman, London (1978)

    MATH  Google Scholar 

  33. Spiridonov, V., Zhedanov, A.: Discrete-time Volterra chain and classical orthogonal polynomials. J. Phys. A, Math. Gen. 30, 8727–8737 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  34. Szegő, G.: Orthogonal Polynomials, 4th edn. AMS, Providence (1975)

    Google Scholar 

  35. Veselov, A.P., Shabat, A.B.: Dressing chains and the spectral theory of the Schrödinger operator. Funct. Anal. Appl. 27(2), 81–96 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  36. Vinet, L., Zhedanov, A.: A “missing” family of classical orthogonal polynomials. J. Phys. A, Math. Theor. 44, 085201 (2011). arXiv:1011.1669v2

    Article  MathSciNet  Google Scholar 

  37. Vinet, L., Zhedanov, A.: A Bochner theorem for Dunkl polynomials. SIGMA 7, 020 (2011). arXiv:1011.1457v3

    MathSciNet  Google Scholar 

  38. Vinet, L., Zhedanov, A.: A limit q=−1 for big q-Jacobi polynomials. Trans. Am. Math. Soc. (to appear). arXiv:1011.1429v3

  39. Watkins, D.S.: Some perspectives on the eigenvalue problem. SIAM Rev. 35, 430–471 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zhedanov, A.: Rational spectral transformations and orthogonal polynomials. J. Comput. Appl. Math. 85(1), 67–86 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  41. Zhedanov, A.: On some classes of polynomials orthogonal on arcs of the unit circle connected with symmetric orthogonal polynomials on an interval. J. Approx. Theory 94, 73–106 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhedanov, A.: The “classical” Laurent biorthogonal polynomials. J. Comput. Appl. Math. 98, 121–147 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zhedanov, A.: Biorthogonal rational functions and the generalized eigenvalue problem. J. Approx. Theory 101, 303–329 (1999)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are very grateful to the referees and the editor for many valuable comments and suggestions which helped to improve the manuscript and also gave us some ideas for further developments. The research of LV is supported in part by a research grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada.

Author information

Authors and Affiliations

  1. Department of Mathematics MA 3-6, Technische Universität Berlin, Strasse des 17. Juni 136, 10623, Berlin, Germany

    Maxim Derevyagin

  2. Centre de Recherches Mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station, Montréal (Québec), H3C 3J7, Canada

    Luc Vinet

  3. Donetsk Institute for Physics and Technology, 83114, Donetsk, Ukraine

    Alexei Zhedanov

Authors
  1. Maxim Derevyagin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luc Vinet
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alexei Zhedanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Luc Vinet.

Additional information

Communicated by Tom H. Koornwinder.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Derevyagin, M., Vinet, L. & Zhedanov, A. CMV Matrices and Little and Big −1 Jacobi Polynomials. Constr Approx 36, 513–535 (2012). https://doi.org/10.1007/s00365-012-9164-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00365-012-9164-0

Keywords

Mathematics Subject Classification

Search

Navigation