Quadrature rules from finite orthogonality relations for Bernstein-Szegö polynomials
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- by J. F. van Diejen and E. Emsiz PDF
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Abstract:
We glue two families of Bernstein-Szegö polynomials to construct the eigenbasis of an associated finite-dimensional Jacobi matrix. This gives rise to finite orthogonality relations for this composite eigenbasis of Bernstein-Szegö polynomials. As an application, a number of Gauss-like quadrature rules are derived for the exact integration of rational functions with prescribed poles against the Chebyshev weight functions.References
- Richard Askey, Positive quadrature methods and positive polynomial sums, Approximation theory, V (College Station, Tex., 1986) Academic Press, Boston, MA, 1986, pp. 1–29. MR 903680
- Richard Askey and James Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319, iv+55. MR 783216, DOI 10.1090/memo/0319
- Adhemar Bultheel, Ruymán Cruz-Barroso, Karl Deckers, and Pablo González-Vera, Rational Szegő quadratures associated with Chebyshev weight functions, Math. Comp. 78 (2009), no. 266, 1031–1059. MR 2476569, DOI 10.1090/S0025-5718-08-02208-4
- David Damanik and Barry Simon, Jost functions and Jost solutions for Jacobi matrices. II. Decay and analyticity, Int. Math. Res. Not. , posted on (2006), Art. ID 19396, 32. MR 2219207, DOI 10.1155/IMRN/2006/19396
- L. Daruis, P. González-Vera, and M. Jiménez Paiz, Quadrature formulas associated with rational modifications of the Chebyshev weight functions, Comput. Math. Appl. 51 (2006), no. 3-4, 419–430. MR 2207429, DOI 10.1016/j.camwa.2005.10.004
- Philip J. Davis and Philip Rabinowitz, Methods of numerical integration, 2nd ed., Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. MR 760629
- J. F. van Diejen and E. Emsiz, Discrete Fourier transform associated with generalized Schur polynomials, Proc. Amer. Math. Soc. 146 (2018), no. 8, 3459–3472. MR 3803671, DOI 10.1090/proc/14036
- J. F. van Diejen and E. Emsiz, Exact cubature rules for symmetric functions, Math. Comp., DOI 10.1090/mcom/3380.
- Abdelkrim Ezzirani and Allal Guessab, A fast algorithm for Gaussian type quadrature formulae with mixed boundary conditions and some lumped mass spectral approximations, Math. Comp. 68 (1999), no. 225, 217–248. MR 1604332, DOI 10.1090/S0025-5718-99-01001-7
- Walter Gautschi, A survey of Gauss-Christoffel quadrature formulae, E. B. Christoffel (Aachen/Monschau, 1979) Birkhäuser, Basel-Boston, Mass., 1981, pp. 72–147. MR 661060
- Walter Gautschi, Generalized Gauss-Radau and Gauss-Lobatto formulae, BIT 44 (2004), no. 4, 711–720. MR 2211041, DOI 10.1007/s10543-004-3812-0
- Jeffrey S. Geronimo and Plamen Iliev, Bernstein-Szegő measures, Banach algebras, and scattering theory, Trans. Amer. Math. Soc. 369 (2017), no. 8, 5581–5600. MR 3646771, DOI 10.1090/tran/6841
- Zinoviy Grinshpun, On oscillatory properties of the Bernstein-Szegő orthogonal polynomials, J. Math. Anal. Appl. 272 (2002), no. 1, 349–361. MR 1930719, DOI 10.1016/S0022-247X(02)00169-5
- Zinoviy Grinshpun, The Christoffel function of Bernstein-Szegő orthogonal polynomials, Far East J. Math. Sci. (FJMS) 26 (2007), no. 2, 257–274. MR 2360206
- Hédi Joulak and Bernhard Beckermann, On Gautschi’s conjecture for generalized Gauss-Radau and Gauss-Lobatto formulae, J. Comput. Appl. Math. 233 (2009), no. 3, 768–774. MR 2583015, DOI 10.1016/j.cam.2009.02.083
- Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw, Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. With a foreword by Tom H. Koornwinder. MR 2656096, DOI 10.1007/978-3-642-05014-5
- C. A. Micchelli and T. J. Rivlin, Numerical integration rules near Gaussian quadrature, Israel J. Math. 16 (1973), 287–299. MR 366003, DOI 10.1007/BF02756708
- Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark (eds.), NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2723248
- Franz Peherstorfer, Linear combinations of orthogonal polynomials generating positive quadrature formulas, Math. Comp. 55 (1990), no. 191, 231–241. MR 1023052, DOI 10.1090/S0025-5718-1990-1023052-9
- Guergana Petrova, Generalized Gauss-Radau and Gauss-Lobatto formulas with Jacobi weight functions, BIT 57 (2017), no. 1, 191–206. MR 3608316, DOI 10.1007/s10543-016-0627-8
- Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517
- Yuan Xu, A characterization of positive quadrature formulae, Math. Comp. 62 (1994), no. 206, 703–718. MR 1223234, DOI 10.1090/S0025-5718-1994-1223234-0
Additional Information
- J. F. van Diejen
- Affiliation: Instituto de Matemática y Física, Universidad de Talca, Casilla 747, Talca, Chile
- MR Author ID: 306808
- ORCID: 0000-0002-5410-8717
- Email: diejen@inst-mat.utalca.cl
- E. Emsiz
- Affiliation: Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile
- MR Author ID: 781405
- Email: eemsiz@mat.uc.cl
- Received by editor(s): January 3, 2018
- Received by editor(s) in revised form: April 5, 2018
- Published electronically: August 14, 2018
- Additional Notes: This work was supported in part by the Fondo Nacional de Desarrollo Científico y Tecnológico (FONDECYT) Grants # 1170179 and # 1181046
- Communicated by: Mourad Ismail
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 5333-5347
- MSC (2010): Primary 65D32; Secondary 33C47, 33D45, 47B36
- DOI: https://doi.org/10.1090/proc/14186
- MathSciNet review: 3866872