Mathematics of Computation

Author(s): Sven Ehrich; Giuseppe Mastroianni.
Journal: Math. Comp. 66 (1997), 311-331.
MSC (1991): Primary 42A05, 65D05
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Abstract: Bounds are proved for the Stieltjes polynomial $E_{n+1}$ , and lower bounds are proved for the distances of consecutive zeros of the Stieltjes polynomials and the Legendre polynomials $\P$ . This sharpens a known interlacing result of Szegö. As a byproduct, bounds are obtained for the Geronimus polynomials $G_n$

. Applying these results, convergence theorems are proved for the Lagrange interpolation process with respect to the zeros of $E_{n+1}$ , and for the extended Lagrange interpolation process with respect to the zeros of $\P E_{n+1}$ in the uniform and weighted $L^p$

norms. The corresponding Lebesgue constants are of optimal order.

1.: P. BARRUCAND: Intégration Numérique, Abscisses de Kronrod-Patterson et Polynômes de Szegö. C.R. Acad. Sci. Paris, Ser. A, 270, 1970, 147-158. MR 42:2657
2.: H. BRASS: Quadraturverfahren. Vandenhoeck & Ruprecht, Göttingen, 1977. MR 56:1675
3.: G. CRISCUOLO AND G. MASTROIANNI: Fourier and Lagrange Operators in some weighted Sobolev-type spaces. Acta Math. Scient. Szeged. 60, 1995, 131-148. MR 96e:41006
4.: G. CRISCUOLO, G. MASTROIANNI AND P. NEVAI: Mean convergence of the derivatives of extended Lagrange interpolation with additional nodes, Math. Nachr. 163, 1993, 73-92. MR 94g:65011
5.: G. CRISCUOLO, G. MASTROIANNI AND D. OCCORSIO: Convergence of extended Lagrange interpolation, Math. Comp. 55, 1990, 197-212. MR 91c:65008
6.: G. CRISCUOLO, G. MASTROIANNI AND P. VÉRTESI: Pointwise simultaneous convergence of the extended Lagrange interpolation with additional knots, Math. Comp. 59, 1992, 515-531. MR 93a:41003
7.: G. CRISCUOLO, L. SCUDERI: Error bounds for product quadrature rules in weighted norm, Calcolo, 31, 1994, 73-93. CMP 96:03
8.: B. DELLA VECCHIA, G. MASTROIANNI AND P. VÉRTESI: Boundedness of Lagrange and Hermite operators. Preprint.
9.: L. DURAND: Nicholson-type integrals for products of Gegenbauer functions and related topics. In: R. Askey, Ed., Theory and Application of Special Functions, Academic Press, New York, 1975, 353-374. MR 53:13679
10.: S. EHRICH: Asymptotic Properties of Stieltjes Polynomials and Gauss-Kronrod Quadrature Formulae, J. Approx. Theory 82, 1995, 287-303. CMP 95:16
11.: K.-J. FÖRSTER: A Comparison Theorem for Linear Functionals and its application in Quadrature. In Numerical Integration, Proc. Conf. Oberwolfach, G. Hämmerlin, ed., ISNM 57, Birkhäuser, Basel 1982, 66-76.
12.: K.-J. FÖRSTER: Schranken für die Varianz und die Gewichte von Quadraturformeln. Habilitationsschrift, Tech. Univ. Braunschweig, Germany, 1987.
13.: K.-J. FÖRSTER: Inequalities for Legendre polynomials and application to quadrature. J. Comp. Appl. Math. 33, 1981, 85-95.
14.: K.-J. FÖRSTER AND K. PETRAS: On Estimates for the weights in Gaussian Quadrature in the ultraspherical case. Math. Comp. 55, 1990, 243-264. MR 91d:65043
15.: W. GAUTSCHI: Gauss-Kronrod Quadrature - A Survey. In Numerical Methods and Approximation Theory III, G.V. Milovanovi\'{c}, ed., Ni[??]s 1988, 39-66. MR 89k:41035
16.: W. GAUTSCHI AND S.E. NOTARIS: An Algebraic and Numerical Study of Gauss-Kronrod Quadrature Formulae for Jacobi Weight Functions. Math. Comp. 51, 1988, 231-248. MR 89f:65031
17.: W. GAUTSCHI AND T.J. RIVLIN: A Family of Gauss-Kronrod Quadrature Formulae. Math. Comp. 51, 1988, 749-754. MR 89m:65029
18.: J. GERONIMUS: On a set of Polynomials. Annals Math. 31, 1930, 681-686.
19.: D. LUBINSKY, A. MÁTÉ AND P. NEVAI: Quadrature sums involving th powers of polynomials. SIAM J. Math. Anal. 18, 1987, 531-544. MR 89h:41058
20.: G. MASTROIANNI: Uniform Convergence of derivatives of Lagrange interpolation. J. Comp. Appl. Math. 43, 1992, 37-51. MR 93j:41006
21.: G. MASTROIANNI: Approximation of functions by extended Lagrange interpolation. In Approximation and Computation, A Festschrift In Honor of Walter Gautschi, R.V.M. Zahar, ed., ISNM 119, Birkhäuser, 1995, 409-420.
22.: G. MASTROIANNI AND P. NEVAI: Mean Convergence of Lagrange Interpolation. J. Comp. Appl. Math. 34, 1991, 385-396. MR 92b:41007
23.: G. MASTROIANNI AND P. VÉRTESI: Weighted error of Lagrange interpolation, to appear in J. Approx. Theory.
24.: G. MASTROIANNI AND P. VÉRTESI: Error estimates of product quadrature rules. In Numerical Integration IV, Proc. Conf. Oberwolfach, H. Brass and G. Hämmerlin, eds., ISNM 112, Birkhäuser, 1993, 241-252.
25.: G. MONEGATO: A Note on Extended Gaussian Quadrature Rules. Math. Comp. 30, 1976, 812-817. MR 55:13746
26.: G. MONEGATO: Positivity of Weights of Extended Gauss-Legendre Quadrature Rules. Math. Comp. 32, 1978, 243-245. MR 56:17009
27.: G. MONEGATO: An Overview of Results and Questions Related to Kronrod Schemes. In Numerische Integration, Proc. Conf. Oberwolfach, G. Hämmerlin, ed., ISNM 57, Birkhäuser 1979. MR 81m:65033
28.: G. MONEGATO: Stieltjes Polynomials and Related Quadrature Rules. SIAM Review 24, 1982, 137-158. MR 83d:65067
29.: B. MUCKENHOUPT AND R. WHEEDEN: Two weight function norm inequalities for the Hardy-Littlewood maximal function and the Hilbert transform. Studia Math. 55, 1976, 279-294. MR 54:5720
30.: P. NEVAI: Mean Convergence of Lagrange Interpolation, III. Trans. Amer. Math. Soc. 282, 1984, 669-698. MR 85c:41009
31.: P. NEVAI: Hilbert Transform and Lagrange Interpolation. J. Approx. Theory 60, 1990, 360-363. MR 91a:42008
32.: D. OCCORSIO: Una buona matrice di nodi, Calcolo 30, 1993, 107-126. MR 96f:41005
33.: D. OCCORSIO: Convergence of extended Lagrange interpolation in weighted , Calcolo, 31, 1994, 47-61. CMP 96:03
34.: F. PEHERSTORFER: On the Asymptotic Behaviour of Functions of the Second Kind and Stieltjes Polynomials and on the Gauss-Kronrod Quadrature Formulas. J. Approx. Theory 70, 1992, 156-190. MR 93h:42020
35.: T.J. RIVLIN: Chebyshev Polynomials. Wiley-Interscience, 1990. MR 92a:41016
36.: G. SZEGÖ: Über gewisse orthogonale Polynome, die zu einer oszillierenden Belegungsfunktion gehören. Math. Ann. 110, 1934, 501-513.
37.: G. SZEGÖ: Orthogonal Polynomials. AMS Colloq. Publ. 23, Providence, RI 1975. MR 51:8724
38.: Y. Xu: Mean Convergence of Generalized Jacobi Series and interpolating polynomials, I. J. Approx. Theory 72, 1993, 237-251. MR 94d:41010

Sven Ehrich
Affiliation: Universität Hildesheim, Institut für Mathematik, D--31141 Hildesheim, Germany
Email: ehrich@informatik.uni-hildesheim.de

Giuseppe Mastroianni
Affiliation: Università degli Studi della Basilicata, Dipartimento di Matematica, I--85100 Potenza, Italy
Email: mastroianni@pzvx85.cineca.it

DOI: 10.1090/S0025-5718-97-00808-9
PII: S 0025-5718(97)00808-9
Keywords: Stieltjes polynomials, Lagrange interpolation, extended Lagrange interpolation, convergence
Received by editor(s): June 20, 1995
Received by editor(s) in revised form: December 4, 1995
Copyright of article: Copyright 1997, American Mathematical Society

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