Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-21T02:08:29.665Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic Monotonicity of the Relative Extrema of Jacobi Polynomials

Published online by Cambridge University Press:  20 November 2018

R. Wong  and
J.-M. Zhang
R. Wong
Affiliation:
Department of Applied Mathematics, University of Manitoba Winnipeg, ManitobaR3T 2N2
J.-M. Zhang
Affiliation:
Department of Applied Mathematics, Tsinghua University, Beijing China
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

If μk,n(α,β) denotes the relative extrema of the Jacobi polynomial P(α,β)n(x), ordered so that μk+1,n(α,β) lies to the left of μk,n(α,β), then R. A. Askey has conjectured twenty years ago that for for k = 1,…, n — 1 and n = 1,2,=. In this paper, we give an asymptotic expansion for μk,n(α,β) when k is fixed and n → ∞, which corrects an earlier result of R. Cooper (1950). Furthermore, we show that Askey's conjecture is true at least in the asymptotic sense.

Keywords

33C4541A60Jacobi polynomialszerosrelative extremauniform asymptotic approximation
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 6 , 01 December 1994 , pp. 1318 - 1337
Copyright
Copyright © Canadian Mathematical Society 1994

References

1. Askey, R. A., Graphs as an aid to understanding special functions, Asymptotic and Computational Analysis, (ed. R. Wong), Marcel Dekker, New York, 1990, 3–33.Google Scholar
2. Baratella, P. and Gatteschi, L., The bounds for the error terms of an asymptotic approximation of Jacobi polynomials, Orthogonal Polynomials and Their Applications, (eds. M. Alfaro et al), Lecture Notes in Math. 1329, Springer-Verlag, Berlin, New York, 1988, 203–221.Google Scholar
3. Cooper, R., The extremal values of Le gendre polynomials and of certain related functions, Math. Proc. Cambridge Philos. Soc. 46(1950), 549–554.Google Scholar
4. Erdélyi, A., Magnus, A., Oberhettinger, F. and Tricomi, F., Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York, 1953.Google Scholar
5. Frenzen, C. L. and Wong, R., A uniform asymptotic expansion of the Jacobi polynomials with error bounds, Canad. J. Math. 37(1985), 979–1007.Google Scholar
6. Frenzen, C. L., Asymptotic expansions of the Lebesgue constants for Jacobi series, Pacific J. Math. 122(1986), 391–415.Google Scholar
7. Olver, F. W. J., Asymptotics and Special Functions, Academic Press, New York, 1974.Google Scholar
8. Szàsz, O., On the relative extrema of ultraspherical polynomials, Boll. Un. Mat. Ital. (3) 5(1950), 125–127.Google Scholar
9. Szegö, G., Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. 23, Amer. Math. Soc, Providence, 1939, fourth edition, 1975.Google Scholar
10. Szegö, G., On the relative extrema of Le gendre polynomials, Boll. Un. Mat. Ital. (3) 5(1950), 120–121.Google Scholar
11. Todd, J., On the relative extrema of the Laguerre polynomials, Boll. Un. Mat. Ital. (3) 5(1950), 122–125.Google Scholar
12. Wong, R. and Zhang, J.-M., On the relative extrema of the Jacobi polynomials , SIAM J. Math. Anal. 25(1994), 776–811.Google Scholar