A survey on orthogonal matrix polynomials satisfying second order differential equations

Journal of Computational and Applied Mathematics

Volumn 178, Issue 1-2 SPEC. ISS., 2005, Pages 169-190

  Durán, Antonio J ^a   Grünbaum, F Alberto ^b  

^a UNIVERSITY OF SEVILLE   (Spain)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Index of determinacy; Orthogonal matrix polynomials; Orthogonal polynomials

Indexed keywords

CALCULATIONS; DIFFERENTIAL EQUATIONS; FUNCTIONS; MATHEMATICAL OPERATORS; MATRIX ALGEBRA; THEOREM PROVING;

INDEX OF DETERMINACY; ORTHOGONAL MATRIX POLYNOMIALS; ORTHOGONAL POLYNOMIALS; SECOND ORDER DIFFERENTIAL EQUATIONS;

POLYNOMIALS;

EID: 14844346986     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2004.05.023     Document Type: Article

References (42)

1. S. Basu N.K. Bose Matrix Stieltjes series and network models SIAM J. Math. Anal. 14 1983 209-222
2. Ju.M. Berezanskii Expansions in eigenfunctions of selfadjoint operators Transl. Math. Monographs AMS 17 1968
3. S. Bochner S. Über Sturm-Liouvillesche polynomsysteme Math. Z. 29 1929 730-736
4. M. Cantero, L. Moral, L. Velazquez, Differential properties of matrix orthogonal polynomials, J. Comput. Appl. Math., to appear.
5. H. Dette J. Studden Matrix measures, moment spaces and Favard's theorem for the interval [0 , 1] and [0 ,∞) Linear Algebra Appl. 345 2002 169-193
6. J.J. Duistermaat F.A. Grünbaum Differential equations in the spectral parameter Comm. Math. Phys. 103 1986 177-240
7. A.J. Duran A generalization of Favard's theorem for polynomials satisfying a recurrence relation J. Approx. Theory 74 1994 260-275
8. A.J. Duran On orthogonal polynomials with respect to a positive definite matrix of measures Can. J. Math. 47 1995 88-112
9. A.J. Duran Markov's theorem for orthogonal matrix polynomials Can. J. Math. 48 1996 1180-1195
10. A.J. Duran Matrix inner product having a matrix symmetric second order differential operator Rocky Mount. J. Math. 27 1997 585-600
11. A.J. Duran Ratio asymptotics for orthogonal matrix polynomials J. Approx. Theory 100 1999 304-344
12. A.J. Duran E. Daneri-Vias Ratio asymptotics for orthogonal matrix polynomials with unbounded recurrence coefficients J. Approx. Theory 110 2001 1-17
13. A.J. Duran E. Daneri-Vias Weak convergence for orthogonal matrix polynomials Indag. Math. 13 2002 47-62
14. A.J. Duran E. Defez Orthogonal matrix polynomials and quadrature formulas Linear Algebra Appl. 345 2002 71-84
15. A.J. Duran F.A. Grünbaum Orthogonal matrix polynomials satisfying second order differential equations Int. Math. Res. Notes 10 2004 461-484
16. A.J. Duran, F.A. Grünbaum, Orthogonal matrix polynomials, scalar type Rodrigues' formulas and Pearson equations, submitted for publication.
17. A.J. Duran, F.A. Grünbaum, Structural formulas for orthgonal matrix polynomials satisfying second order differential equations, I, Constr. Approx., to appear.
18. A.J. Duran, F.A. Grünbaum, A characterization for a class of weight matrices with orthogonal matrix polynomials satisfying second order differential equations, submitted for publication.
19. A.J. Duran P. Lopez-Rodriguez Orthogonal matrix polynomials: Zeros and Blumenthal's theorem J. Approx. Theory 84 1996 96-118
20. A.J. Duran P. Lopez-Rodriguez E.B. Saff Zero asymptotic behaviour for orthogonal matrix polynomials J. d'Analyse Math. 78 1999 37-60
21. A.J. Duran B. Polo Gauss quadrature formulae for orthogonal matrix polynomials Linear Algebra Appl. 355 2002 119-146
22. A.J. Duran W. Van Assche Orthogonal matrix polynomials and higher order recurrence relations Linear Algebra Appl. 219 1995 261-280
23. J.S. Geronimo Scattering theory and matrix orthogonal polynomials on the real line Circuits Systems Signal Process. 1 1982 471-495
24. F.A. Grünbaum A new property of reproducing kernels for classical orthogonal polynomials J. Math. Anal. Appl. 95 2 1983 491-500
25. F.A. Grünbaum Time-band limiting and the bispectral problem Comm. Pure Appl. Math. 47 3 1994 307-328
26. F.A. Grünbaum Matrix valued Jacobi polynomials Bull. Sci. Math. 127 3 2003 207-214
27. F.A. Grünbaum, L. Haine, A theorem of Bochner revisited, in: A.S. Fokas, I.M. Gelfand (Eds.), Algebraic Aspects of Integrable Systems, Programming in Nonlinear Differential Equations, vol. 26, Birkhäuser, Boston, 1997, pp. 143-172.
28. F.A. Grünbaum P. Iliev A noncommutative version of the bispectral problem J. Comput. Appl. Math. 161 2003 99-118
29. F.A. Grünbaum L. Longhi M. Perlstadt Differential operators commuting with finite convolution integral operators: Some non-Abelian examples SIAM J. Appl. Math. 42 1982 941-955
30. F.A. Grünbaum, L. Miranian, The magic of the prolate spheroidal wave functions in various setups, SPIE Proceedings, San Diego Meeting, July 2001.
31. F.A. Grünbaum I. Pacharoni J.A. Tirao A matrix valued solution to Bochner's problem J. Phys. A: Math. Gen. 34 2001 10647-10656
32. F.A. Grünbaum I. Pacharoni J.A. Tirao Matrix valued spherical functions associated to the complex projective plane J. Funct. Anal. 188 2002 350-441
33. 2 (ℂ) D. Healy D. Rockmore Modern Signal Processing vol. 46 2003 MSRI Publication 147-160
34. F.A. Grünbaum I. Pacharoni J.A. Tirao Matrix valued orthogonal polynomials of the Jacobi type Indag. Math. 14 2003 353-366
35. L. Jodar R. Company E. Navarro Laguerre matrix polynomials and systems of second order differential equations Appl. Numer. Math. 15 1994 53-63
36. M.G. Krein Infinite J-matrices and a matrix moment problem Dokl. Akad. Nauk SSSR 69 2 1949 125-128
37. M.G. Krein, Fundamental aspects of the representation theory of hermitian operators with deficiency index (m, m), AMS Translations, Series 2, vol. 97, Providence, Rhode Island, 1971, pp. 75-143.
38. F. Marcellan M.A. Piñar H.O. Yakhlef Relative asymptotics for orthogonal matrix polynomials with convergent recurrence coefficients J. Approx. Theory 111 2001 1-30
39. A. Sinap W. van Assche Orthogonal matrix polynomials and applications J. Comput. Appl. Math. 66 1996 27-52
40. D. Slepian Some comments on Fourier analysis, uncertainty and modeling SIAM Rev. 25 3 1983
41. J. Tirao Spherical functions Rev. de la Unión Matem. Argentina 28 1977 75-98
42. J. Tirao The matrix valued hypergeometric equation PNAS 100 4 2003 8138-8141