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

Citation
F. Peherstorfer et K. Petras, Ultraspherical Gauss-Kronrod quadrature is not possible for lambda > 3, SIAM J NUM, 37(3), 2000, pp. 927-948
Citations number
24
Categorie Soggetti
Mathematics
Journal title
SIAM JOURNAL ON NUMERICAL ANALYSIS
ISSN journal
00361429 → ACNP
Volume
37
Issue
3
Year of publication
2000
Pages
927 - 948
Database
ISI
SICI code
0036-1429(20000309)37:3<927:UGQINP>2.0.ZU;2-K
Abstract
With the help of a new representation of the Stieltjes polynomial it is sho wn by using Bessel functions that the Stieltjes polynomial with respect to the ultraspherical weight function with parameter lambda has only few real zeros for lambda > 3 and sufficiently large n. Since the nodes of the Gauss -Kronrod quadrature formulae subdivide into the zeros of the Stieltjes poly nomial and the Gaussian nodes, it follows immediately that Gauss-Kronrod qu adrature is not possible for lambda > 3. On the other hand, for lambda = 3 and sufficiently large n, even partially positive Gauss-Kronrod quadrature is possible.