Ultraspherical Gauss-Kronrod quadrature is not possible for λ > 3

SIAM Journal on Numerical Analysis

Volumn 37, Issue 3, 2000, Pages 927-948

  Peherstorfer, Franz ^a   Petras, Knut ^b  

^a JOHANNES KEPLER UNIVERSITY LINZ   (Austria)

^b TECHNICAL UNIVERSITY OF BRAUNSCHWEIG   (Germany)

Author keywords

Bessel functions; Gauss Kronrod quadrature; Orthogonal polynomials; Stieltjes polynomials

Indexed keywords

EID: 0010102013     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142998327744     Document Type: Article

References (24)

1. M. ABRAMOWITZ AND I. STEGUN, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1964.
2. P. BARRUCAND, Integration numérique, abscisse de Kronrod-Patterson et polynôme de Szegö, C. R. Acad. Sci. Paris Sér. A-B, 270 (1970), pp. 336-338.
3. D. J. DICKINSON, On certain polynomials associated with orthogonal polynomials, Boll. Un. Mat. (3) Ital., 13 (1958), pp. 116-124.
4. S. EHRICH, Asymptotic properties of Stieltjes polynomials and Gauss-Kronrod quadrature formulae, J. Approx. Theory, 82 (1995), pp. 287-303.
5. S. EHRICH, Asymptotic behaviour of Stieltjes polynomials for ultraspherical weight functions, J. Comput. Appl. Math., 65 (1995), pp. 135-144.
6. D. ELLIOT, Uniform asymptotic expansions of the Jacobi polynomials and an associated function, Math. Comp., 25 (1971), pp. 309-315.
7. G. FREUD, Orthogonal Polynomials, Pergamon Press, Oxford, 1971.
8. W. GAUTSCHI AND S. NOTARIS, An algebraic study of Gauss-Kronrod quadrature formulae for Jacobi weight functions, Math. Comp., 51 (1988), pp. 231-248.
9. I. S. GRADSHTEYN AND I. M. RYZHIK, Table of Series, Integrals and Products, Academic Press, New York, 1965.
10. A. S. KRONROD, Nodes and Weights for Quadrature Formulae. Sixteen-Place Tables. Nauka, Moscow, 1964 (in Russian); Consultants Bureau, New York, 1965 (in English).
11. G. MONEGATO, Positivity of the weights of extended Gauss-Legendre quadrature rules, Math. Comp., 32 (1978), pp. 243-245.
12. G. MONEGATO, Stieltjes polynomials and related quadrature rules, SIAM Rev., 24 (1982), pp. 137-158.
13. P. NEVAI, A new class of orthogonal polynomials, Proc. Amer. Math. Soc., 91 (1984), pp. 409-415.
14. S. NOTARIS, An Overview of Results on the Existence or Nonexistence and the Error Term of Gauss-Kronrod Quadrature Formulae, manuscript.
15. F. W. J. OLVER, The asymptotic solution of linear differential equations of the second order in a domain containing one transition point, Philos. Trans. Roy. Soc. London, Ser. A, 249 (1956), pp. 65-97.
16. F. PEHERSTORFER, On the asymptotic behaviour of functions of the second kind and Stieltjes polynomials and on Gauss-Kronrod quadrature formulas, J. Approx. Theory, 70 (1992), pp. 156-190.
17. F. PEHERSTORFER, Finite perturbations of orthogonal polynomials, J. Comput. Appl. Math., 44 (1992), pp. 275-302.
18. F. PEHERSTORFER, Stieltjes polynomials and functions of the second kind, J. Comput. Appl. Math., 65 (1995), pp. 319-338.
19. P. RABINOWITZ, Gauss-Kronrod integration rules for Cauchy principle value integrals, Math. Comp., 41 (1983), pp. 63-78.
20. T. J. STIELTJES, Correspondance d'Hermite et de Stieltjes, Vol. II, Gauthier-Villars, Paris, 1905.
21. G. SZEGÖ, Über gewisse Polynome, die zu einer oszillierenden Belegungsfunktion gehören, Math. Ann., 110 (1935), pp. 501-513.
22. G. SZEGÖ, Orthogonal Polynomials, A.M.S., Providence, RI 1939.
23. W. VAN ASSCHE, Orthogonal polynomials, associated polynomials and functions of the second kind, J. Comput. Appl. Math., 37 (1991), pp. 237-249.
24. G. N. WATSON, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, UK, 1944.