# On Mean Convergence of Lagrange-Kronrod Interpolation

• Shikang Li
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 119)

## Abstract

It is known that Lagrange interpolation to any continuous function on [-1, 1] at the zeros of an orthogonal polynomial converges in the mean. In this paper we study mean convergence of Lagrange-Kronrod interpolation where the nodes are the n zeros of an orthogonal polynomial of degree n and the n + 1 zeros of the associated Stieltjes polynomial of degree n + 1. A sufficient condition for mean convergence, due to P. Erdös and P. Turán, is shown to hold for the first-, the third- and fourth-kind Chebyshev weight functions.

## Keywords

Weight Function Orthogonal Polynomial Quadrature Formula Lagrange Interpolation Lagrange Interpolation Polynomial
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## References

1. [1]
Bellen A. Alcuni problemi aperti sulla convergenza in media dell’interpolazione Lagrangiana estesa. Rend. 1st. Mat Univ. Trieste, 20: 1–9, 1988.
2. [2]
Davis P. J., Rabinowitz P. Methods of numerical integration. Academic Press, Orlando, FI., 1984.
3. [3]
Erdös P., Turàn P. On interpolation I. Ann. of Math., 38: 142–155, 1937.
4. [4]
Gautschi W. Gauss-Kronrod quadrature-A survey. In Numerical Methods and Approximation Theory III, pages 39–66. Faculty of Electronic Engineering, Univ. Nis, Nis, 1988. G. V. Milovanović, ed.Google Scholar
5. [5]
Gautschi W. On mean convergence of extended Lagrange interpolation. J. Comput. Appl Math., 43: 19–35, 1992.
6. [6]
Kronrod A. S. Nodes and weights for quadrature formulae. Sixteen-place tables (Russian). Izdat. “Nauka”, Moscow, 1964.Google Scholar
7. [7]
Kronrod A. S. Integration with control of accuracy (Russian). Dokl. Akad. Nauk SSSR, 154: 283–286, 1964.
8. [8]
Monegato G. Stieltjes polynomials and related quadrature rules. SIAM Review, 24: 137–158, 1982.
9. [9]
Szegö G. Über gewisse orthogonale Polynome, die zu einer oszillierenden Belegungsfunktion gehören. Math. Ann., 110: 501–513, 1934.