On the convergence of multipoint Padé-type approximants and quadrature formulas associated with the unit circle

Numerical Algorithms

Volumn 13, Issue 2, 1996, Pages 321-344

  Bultheel, A ^a   Gonzalez Vera P ^b   Hendriksen, E ^c   Njastad O ^d  

^a UNIVERSITY OF LEUVEN   (Belgium)

^b UNIVERSIDAD DE LA LAGUNA   (Spain)

^c UNIVERSITY OF AMSTERDAM   (Netherlands)

^d NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Norway)

Author keywords

Multipoint Pad type approximation; Orthogonal rational functions; Quadrature formula

Indexed keywords

EID: 21444448688     PISSN: 10171398     EISSN: None     Source Type: Journal    
DOI: 10.1007/bf02207699     Document Type: Article

References (24)

1. N. I. Achieser, Theory of Approximation (Frederick Ungar, New York, 1956).
2. A. Bultheel, P. González-Vera, E. Hendriksen and O. Njåstad, Orthogonal rational functions and quadrature on the unit circle, Numer. Algorithms 3 (1992) 105-116.
3. A. Bultheel, P. González-Vera, E. Hendriksen and O. Njåstad, Orthogonality and boundary interpolation, in: Nonlinear Numerical Methods and Rational Approximation II, ed. A. M. Cuyt (Kluwer, 1994) pp. 37-48.
4. A. Bultheel, P. González-Vera, E. Hendriksen and O. Njåstad, Quadrature formulas on the unit circle and two-point Padé approximation, in: Nonlinear Numerical Methods and Rational Approximation II, ed. A. M. Cuyt (Kluwer, 1994) pp. 303-318.
5. A. Bultheel, P. González-Vera, E. Hendriksen and O. Njåstad, Quadrature formulas on the unit circle based on rational functions, J. Comput. Appl. Math. 50 (1994) 159-170.
6. E. W. Cheney, Introduction to Approximation Theory (McGraw Hill, 1966).
7. M. M. Djrbashian, A survey on the theory of orthogonal systems and some open problems, in: Orthogonal Polynomials: Theory and Practice, ed. P. Nevai, Mathematical and Physical Sciences, Series C 294 (NATO-ASI, Kluwer, Boston, 1990) pp. 135-146,
8. W. Gautschi, Gauss-type quadrature rules for rational functions, in: Numerical Integration IV, eds. H. Brass and G. Hammerlin, Internat. Ser. Numer. Math. 112 (1993) pp. 111-130,
9. E. Godoy and F. Marcellan, Orthogonal polynomials and rational modification of measures, Canad. J. Math. 45 (1993) 930-943.
10. A. A. Gonchar and E. A. Rakhmanov, Equilibrium measure and the distribution of zeros of extremal polynomials, Math. USSR-Sb. 53 (1986) 119-130.
11. P. González-Vera, M. Jiménez-Páiz, G. López-Lagomasino and R. Orive, On the convergence of quadrature formulas connected with multipoint Padé-type approximants, J. Math. Anal. Appl. 202 (1996) 747-775.
12. P. González-Vera, G. López-Lagomasino, R. Orive and J. C. Santos-Leon, On the convergence of quadrature formulas for complex weight functions, J. Math. Anal. Appl. 189 (1995) 514-532.
13. P. González-Vera, O. Njåstad and J.C. Santos-Leon, Some numerical results about quadrature formulas on the unit circle, Adv. Comput. Math. 5 (1996) 297-328.
14. J. Illan and G. López-Lagomasino, A note on generalized quadrature formulas of Gauss-Jacobi type, in: Constructive Theory of Functions '84, Sofia (1984) pp. 513-518.
15. W. B. Jones, O. Njåstad and W. J. Thron, Moment theory, orthogonal polynomials, quadrature and continued fractions associated with the unit circle, Bull. London Math. Soc. 21 (1989) 113-152.
16. A. Máté, P. Nevai and V. Totik, Strong and weak convergence of orthogonal polynomials, Amer. J. Math. 109 (1987) 239-281.
17. J. Nuttal and C. J. Wherry, Gauss integration for complex weight functions, J. Inst. Math. Appl. 21 (1987) 165-170.
18. E. A. Rakhmanov, On the asymptotics of the ratio of orthogonal polynomials, Mat. Sb. 32 (1977) 199-213.
19. E. A. Rakhmanov, On the asymptotics of the ratio of orthogonal polynomials II, Mat. Sb. 46 (1983) 105-117.
20. H. Stahl and V. Totik, General Orthogonal Polynomials, Encyclopedia of Mathematics and its Applications (Cambridge University Press, 1992).
21. G. Szego, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. 33 (Amer. Math. Soc., Providence, RI, 3rd ed., 1967; 1st ed., 1939).
22. W. Van Assche and I. Vanherwegen, Quadrature formulas based on rational interpolation, Math. Comp. 16 (1993) 765-783.
23. H. Wallin, Potential theory and approximation of analytic functions by rational interpolation, in: Complex Analysis, Joensuu 1978, eds. I. Laine, O. Letho and T. Sorvali, Lecture Notes in Mathematics 747 (Springer, 1979) pp. 434-450.
24. J. L. Walsh, Interpolation and Approximation, Amer. Math. Soc. Colloq. Publ. 20 (Amer. Math. Soc., Providence, RI, 3rd ed., 1960; 1st ed., 1935).