Szegö quadrature formulas for certain Jacobi-type weight functions

Daruis, Leyla; González-Vera, Pablo; Njåstad, Olav

doi:10.1090/S0025-5718-01-01337-0

Szegö quadrature formulas for certain Jacobi-type weight functions
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by Leyla Daruis, Pablo González-Vera and Olav Njåstad PDF

Math. Comp. 71 (2002), 683-701 Request permission

Abstract:

In this paper we are concerned with the estimation of integrals on the unit circle of the form $\int _0^{2\pi }f(e^{i\theta })\omega (\theta )d\theta$ by means of the so-called Szegö quadrature formulas, i.e., formulas of the type $\sum _{j=1}^n\lambda _jf(x_j)$ with distinct nodes on the unit circle, exactly integrating Laurent polynomials in subspaces of dimension as high as possible. When considering certain weight functions $\omega (\theta )$ related to the Jacobi functions for the interval $[-1,1],$ nodes $\{x_j\}_{j=1}^n$ and weights $\{\lambda _j\}_{j=1}^n$ in Szegö quadrature formulas are explicitly deduced. Illustrative numerical examples are also given.

References

N. I. Akhiezer, The classical moment problem and some related questions in analysis, Hafner Publishing Co., New York, 1965. Translated by N. Kemmer. MR 0184042
Adhemar Bultheel, Algorithms to compute the reflection coefficients of digital filters, Numerical methods of approximation theory, Vol. 7 (Oberwolfach, 1983) Internat. Schriftenreihe Numer. Math., vol. 67, Birkhäuser, Basel, 1984, pp. 33–50. MR 757007
L. Daruis and P. González-Vera. Szegö polynomials and quadrature formulas on the unit circle. Applied Numerical Mathematics 36 (2001) 79–112.
Walter Gautschi and Gradimir V. Milovanović, Polynomials orthogonal on the semicircle, J. Approx. Theory 46 (1986), no. 3, 230–250. MR 840394, DOI 10.1016/0021-9045(86)90064-X
P. González-Vera, J. C. Santos-León, and O. Njåstad, Some results about numerical quadrature on the unit circle, Adv. Comput. Math. 5 (1996), no. 4, 297–328. MR 1414284, DOI 10.1007/BF02124749
William B. Jones, Olav Njåstad, and W. J. Thron, Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle, Bull. London Math. Soc. 21 (1989), no. 2, 113–152. MR 976057, DOI 10.1112/blms/21.2.113
William B. Jones and Haakon Waadeland, Bounds for remainder terms in Szegő quadrature on the unit circle, Approximation and computation (West Lafayette, IN, 1993) Internat. Ser. Numer. Math., vol. 119, Birkhäuser Boston, Boston, MA, 1994, pp. 325–342. MR 1333626
A. Magnus. Semiclassical orthogonal polynomials on the unit circle. MAPA 3072. Special topics in Approximation Theory. Unpublished Technical Report. (1999)
F. Peherstorfer, A special class of polynomials orthogonal on the unit circle including the associated polynomials, Constr. Approx. 12 (1996), no. 2, 161–185. MR 1393285, DOI 10.1007/s003659900008
Franz Peherstorfer and Robert Steinbauer, Characterization of orthogonal polynomials with respect to a functional, Proceedings of the International Conference on Orthogonality, Moment Problems and Continued Fractions (Delft, 1994), 1995, pp. 339–355. MR 1379142, DOI 10.1016/0377-0427(95)00125-5
Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517