h-Transforms and Orthogonal Polynomials

Abstract

We describe some examples of classical and explicit h-transforms as particular cases of a general mechanism, which is related to the existence of symmetric diffusion operators having orthogonal polynomials as spectral decomposition.

References

  1. 1.
    G.W. Anderson, A. Guionnet, O. Zeitouni, An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics, vol. 118 (Cambridge University Press, Cambridge, 2010)Google Scholar
  2. 2.
    D. Bakry, O. Mazet, Characterization of Markov semigroups on \(\mathbb{R}\) associated to some families of orthogonal polynomials, in Séminaire de Probabilités XXXVII. Lecture Notes in Mathematics, vol. 1832 (Springer, Berlin, 2003), pp. 60–80Google Scholar
  3. 3.
    D. Bakry, M. Zani, Random symmetric matrices on Clifford algebras, in GAFA Seminar, 2011–2013. Lecture notes in Mathematics, vol. 2116 (2013), pp. 1–39Google Scholar
  4. 4.
    D. Bakry, I. Gentil, M. Ledoux, Analysis and Geometry of Markov Diffusion Operators. Grund. Math. Wiss., vol. 348 (Springer, Berlin, 2013)Google Scholar
  5. 5.
    D. Bakry, S. Orevkov, M. Zani, Orthogonal polynomials and diffusion operators (2013) [arXiv/1309.5632]Google Scholar
  6. 6.
    P. Bourgade, M. Yor, Random matrices and the Riemann zeta function, in Journées Élie Cartan 2006, 2007 et 2008. Inst. Élie Cartan, vol. 19 (University of Nancy, Nancy, 2009), pp. 25–40 [MR2792032 (2012j:11180)]Google Scholar
  7. 7.
    P. Bourgade, C.P. Hughes, A. Nikeghbali, M. Yor, The characteristic polynomial of a random unitary matrix: a probabilistic approach. Duke Math. J. 145(1), 45–69 (2008) [MR2451289 (2009j:60011)]Google Scholar
  8. 8.
    I. Cherednik, Double affine Hecke algebras and Macdonald’s conjectures. Ann. Math. 141(1), 191–216 (1995)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    N. Demni, T. Hmidi, Spectral distribution of the free unitary Brownian motion: another approach, in Séminaire de Probabilités XLIV. Lecture Notes in Mathematics, vol. 2046 (Springer, Heidelberg, 2012), pp. 191–206 [MR2953348]Google Scholar
  10. 10.
    N. Demni, T. Hamdi, T. Hmidi, Spectral distribution of the free Jacobi process. Indiana Univ. Math. J. 61(3), 1351–1368 (2012) [MR3071702]Google Scholar
  11. 11.
    J.L. Doob, Conditional Brownian motion and the boundary limits of harmonic functions. Bull. Soc. Math. France 85, 431–458 (1957) [MR0109961 (22 #844)]Google Scholar
  12. 12.
    J.L. Doob, Classical Potential Theory and Its Probabilistic Counterpart. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 262 (Springer, New York, 1984) [MR731258 (85k:31001)]Google Scholar
  13. 13.
    Y. Doumerc, Matrix Jacobi Process. Ph.D. thesis, Université Toulouse 3, 2005Google Scholar
  14. 14.
    C. Dunkl, Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc. 311(1), 167–183 (1989)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    C. Dunkl, Y. Xu, Orthogonal Polynomials of Several Variables. Encyclopedia of Mathematics and Its Applications, vol. 81 (Cambridge University Press, Cambridge, 2001)Google Scholar
  16. 16.
    F.J. Dyson, A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3, 1191–1198 (1962)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    P.I. Etingof, A.A. Kirillov Jr., Macdonald’s polynomials and representations of quantum groups. Math. Res. Lett. 1(3), 279–296 (1994) [MR1302644 (96m:17025)]Google Scholar
  18. 18.
    P.I. Etingof, A.A. Kirillov Jr., On the affine analogue of Jack and Macdonald polynomials. Duke Math. J. 78(2), 229–256 (1995) [MR1333499 (97k:17035)]Google Scholar
  19. 19.
    P.J. Forrester, Log-Gases and Random Matrices. London Mathematical Society Monographs Series, vol. 34 (Princeton University Press, Princeton, 2010)Google Scholar
  20. 20.
    G.J. Heckmann, A remark on the Dunkl differential-difference operators, in Harmonic Analysis on Reductive Groups, ed. by W. Barker, P. Sally, Progress in Mathematics, vol. 101 (Birkhauser, Boston, 1991), pp. 181–191Google Scholar
  21. 21.
    G.J. Heckmann, Dunkl operators, in Séminaire Bourbaki 828, 1996–1997, vol. Astérisque (Société Mathématique de France, Paris, 1997), pp. 223–246Google Scholar
  22. 22.
    T. Koornwinder, Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators I. Nederl. Akad. Wetensch. Proc. Ser. A 77=Indag. Math. 36, 48–58 (1974)Google Scholar
  23. 23.
    I.G. Macdonald, Symmetric Functions and Orthogonal Polynomials. University Lecture Series, vol. 12 (American Mathematical Society, Providence, 1998)Google Scholar
  24. 24.
    I.G. Macdonald, Orthogonal Polynomials Associated with Root Systems. Séminaire Lotharingien de Combinatoire, vol. 45 (Université Louis Pasteur, Strasbourg, 2000)Google Scholar
  25. 25.
    I.G. Macdonald, Affine Hecke Algebras and Orthogonal Polynomials. Cambridge Tracts in Mathematics, vol. 157 (Cambridge University Press, Cambridge, 2003) [MR1976581 (2005b:33021)]Google Scholar
  26. 26.
    R. Mansuy, M. Yor, Aspects of Brownian Motion. Universitext (Springer, Berlin, 2008) [MR2454984 (2010a:60278)]MATHCrossRefGoogle Scholar
  27. 27.
    J. Najnudel, B. Roynette, M. Yor, A Global View of Brownian Penalisations. MSJ Memoirs, vol. 19 (Mathematical Society of Japan, Tokyo, 2009)Google Scholar
  28. 28.
    M. Rösler, Dunkl Operators: Theory and Applications. Orthogonal Polynomials and Special Functions (leuven, 2002). Lecture Notes in Mathematics, vol. 1817 (Springer, Berlin, 2003)Google Scholar
  29. 29.
    B. Roynette, M. Yor, Penalising Brownian Paths. Lecture Notes in Mathematics, vol. 1969 (Springer, Berlin, 2009) [MR2504013] (2010e:60003)Google Scholar
  30. 30.
    M. Sghaier, A note on the Dunkl-classical orthogonal polynomials. Integral Transforms Spec. Funct. 23(10), 753–760 (2012). [MR2980875]Google Scholar
  31. 31.
    E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces. Princeton Mathematical Series, vol. 32 (Princeton University Press, 1971)Google Scholar
  32. 32.
    O. Zribi, Orthogonal polynomials associated with the deltoid curve (2014) [arxiv/1403.7712]Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institut de MathématiquesUniversité P. SabatierToulouseFrance

Personalised recommendations