On Quadrature Convergence of Extended Lagrange Interpolation

Authors:
Walter Gautschi and Shikang Li

Journal:
Math. Comp. **65** (1996), 1249-1256

MSC (1991):
Primary 41A05, 65D32; Secondary 33C45

MathSciNet review:
1344613

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Quadrature convergence of the extended Lagrange interpolant for any continuous function is studied, where the interpolation nodes are the zeros of an orthogonal polynomial of degree and the zeros of the corresponding ``induced'' orthogonal polynomial of degree . It is found that, unlike convergence in the mean, quadrature convergence * does* hold for all four Chebyshev weight functions. This is shown by establishing the positivity of the underlying quadrature rule, whose weights are obtained explicitly. Necessary and sufficient conditions for positivity are also obtained in cases where the nodes and interlace, and the conditions are checked numerically for the Jacobi weight function with parameters and . It is conjectured, in this case, that quadrature convergence holds for .

**1.**C. K. Chui, J. Stöckler, and J. D. Ward,*A Faber series approach to cardinal interpolation*, Math. Comp.**58**(1992), no. 197, 255–273. MR**1106961**, 10.1090/S0025-5718-1992-1106961-3**2.**P. Erdös and P. Turán,*On interpolation I*, Ann. Math.**38**(1937), 142--155.**3.**Walter Gautschi,*On mean convergence of extended Lagrange interpolation*, J. Comput. Appl. Math.**43**(1992), no. 1-2, 19–35. Orthogonal polynomials and numerical methods. MR**1193292**, 10.1016/0377-0427(92)90257-X**4.**W. Gautschi,*Algorithm 726*:*ORTHPOL --- A package of routines for generating orthogonal polynomials and Gauss-type quadrature rules*, ACM Trans. Math. Software**20**(1994), 21--62.**5.**Walter Gautschi and Shikang Li,*A set of orthogonal polynomials induced by a given orthogonal polynomial*, Aequationes Math.**46**(1993), no. 1-2, 174–198. MR**1220730**, 10.1007/BF01834006**6.**G. Pólya,*Über die Konvergenz von Quadraturverfahren*, Math. Z.**37**(1933), 264--286.

Retrieve articles in *Mathematics of Computation of the American Mathematical Society*
with MSC (1991):
41A05,
65D32,
33C45

Retrieve articles in all journals with MSC (1991): 41A05, 65D32, 33C45

Additional Information

**Walter Gautschi**

Affiliation:
Department of Computer Sciences, Purdue University, West Lafayette, Indiana 47907-1398

Email:
wxg@cs.purdue.edu

**Shikang Li**

Affiliation:
Department of Mathematics, Southeastern Louisiana University, Hammond, Louisiana 70402

Email:
kli@selu.edu

DOI:
http://dx.doi.org/10.1090/S0025-5718-96-00731-4

Received by editor(s):
April 20, 1995

Article copyright:
© Copyright 1996
American Mathematical Society