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Harmonics and spherical functions on Grassmann manifolds of rank two and two-variable analogues of Jacobi polynomials

Part of the Lecture Notes in Mathematics book series (LNM,volume 571)

Keywords

  • ORtHOGONAL Polynomial
  • Spherical Function
  • Jacobi Polynomial
  • Zonal Function
  • Grassmann Manifold

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. ASKEY, R., Jacobi polynomials, I. New proofs of Koornwinder's Laplace type integral representation and Bateman's bilinear sum, SIAM J. Math. Anal 5 (1974), 119–124.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. BAVINCK, H., Jacobi series and approximation, Mathematical Centre Tracts, No. 39, Mathematisch Centrum, Amsterdam, 1972.

    MATH  Google Scholar 

  3. BAVINCK, H., Convolution operators for Fourier-Jacobi expansions, in "LInear operators and approximation" (P.L. Butzer, J.-P. Kahane & B. Sz.-Nagy, eds.), p. 371–380, ISNM Vol. 20, Birkhäuser-Verlag, Basel, 1972.

    Google Scholar 

  4. BOCHNER, S., Uber Sturm-Liouvillesche Polynomsysteme, Math. Z. 29 (1929), 730–736.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. ERDÉLYI, A., W. MAGNUS, F. OBERHETTINGER & F.G. TRICOMI, Higher Transcendental Functions, Vol. II, McGraw-Hill, New York, 1953.

    MATH  Google Scholar 

  6. GANGOLLI, R., Positive definite kernels on homogeneous spaces and certain stochastic processes related to Levy's Brownian motion of several parameters, Ann. Inst. H. Poincaré sect. B (N.S.) 3 (1967), 121–226.

    MathSciNet  MATH  Google Scholar 

  7. GELBART, S., A theory of Stiefel harmonics, Trans. Amer. Math. Soc. 192 (1974), 29–50.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. HARISH-CHANDRA, Spherical functions on a semi-simple Lie group, I, Amer. J. Math. 80 (1958), 241–310.

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. HELGASON, S., The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds, Acta Math. 113 (1965), 153–180.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. JAMES, A.T., Normal multivariate analysis and the orthogonal group, Ann. Math. Statist. 25 (1954), 40–75.

    CrossRef  MATH  Google Scholar 

  11. JAMES, A.T. & A.G. CONSTANTINE, Generalized Jacobi polynomials as spherical functions of the Grassmann manifold, Proc. London Math. Soc. (3) 29 (1974), 174–192.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. KOORNWINDER, T.H., Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators, I, II, Nederl. Akad. Wetensch. Proc. Ser. A 77 = Indag. Math. 36 (1974), 59–66.

    MathSciNet  Google Scholar 

  13. KOORWINDER, T.H., Two-variable analogues of the classical orthogonal polynomials, in "Theory and application of special functions" (R. Askey, ed.), pp. 435–495, Academic Press, New York, 1975.

    Google Scholar 

  14. KOORNWINDER, T.H. & I.G. SPRINKHUIZEN, Generalized power series expansions for a class of orthogonal polynomials in two variables, Math. Centrum, Amsterdam, Report TW 155 (1976).

    Google Scholar 

  15. LEVINE, D.A., Systems of singular integral operators on spheres, Trans. Amer. Math. Soc. 144 (1969), 493–521.

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. MAASS, H., Zur Theorie der Kugelfunktionen einer Matrixvariabelen, Math. Ann. 135 (1958), 391–416.

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. SPRINKHUIZEN, I.G., Orthogonal polynomials in two variables. A further analysis of the polynomials orthogonal over a region bounded by two lines and a parabola, Math. Centrum, Amsterdam, Report TW 144 (1974), also to appear in SIAM J. Math. Anal.

    Google Scholar 

  18. STRICHARTZ, R., The explicit Fourier decomposition of L2(SO(n)/SO(n-m)), Canad. J. Math. 27 (1975), 294–310.

    CrossRef  MathSciNet  MATH  Google Scholar 

  19. SZEGÓ, G., Orthogonal polynomials, A.M.S. Colloquium Publications, Vol. 23, American Mathematical Society, Providence, R.I., Third ed., 1967.

    Google Scholar 

  20. TON-THAT, T., Lie group representations and harmonic polynomials of a matrix variable, Trans. Amer. Math. Soc. 216 (1976), 1–46.

    CrossRef  MathSciNet  MATH  Google Scholar 

  21. VRETARE, L., Elementary spherical functions on symmetric spaces, Ark. Mat., to appear.

    Google Scholar 

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Koornwinder, T. (1977). Harmonics and spherical functions on Grassmann manifolds of rank two and two-variable analogues of Jacobi polynomials. In: Schempp, W., Zeller, K. (eds) Constructive Theory of Functions of Several Variables. Lecture Notes in Mathematics, vol 571. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0086570

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  • DOI: https://doi.org/10.1007/BFb0086570

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  • Print ISBN: 978-3-540-08069-5

  • Online ISBN: 978-3-540-37496-1

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