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Necessary and Sufficient Conditions for Mean Convergence of Lagrange Interpolation for Erdős Weights II

Published online by Cambridge University Press:  20 November 2018

S. B. Damelin  and
D. S. Lubinsky
S. B. Damelin
Affiliation:
S. B. Damelin Department of Mathematics University of the Witwatersrand P.O. Wits 2050 Republic of South Africa, e-mail: [email protected]
D. S. Lubinsky
Affiliation:
D. S. Lubinsky Department of Mathematics University of the Witwatersrand P.O. Wits 2050 Republic of South Africa, e-mail: [email protected]
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Abstract

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We complete our investigations of mean convergence of Lagrange interpolation at the zeros of orthogonal polynomials pn(W2, x) for Erdős weights W2 = e-2Q. The archetypal example is Wk,α = exp(—Qk,α), where

α > 1, k ≥ 1, and is the k-th iterated exponential. Following is our main result: Let 1 < p < 4 and α ∊ ℝ Let Ln[f] denote the Lagrange interpolation polynomial to ƒ at the zeros of pn(W2, x) = pn(e-2Q, x). Then for

to hold for every continuous function ƒ:ℝ. —> ℝ satisfying

it is necessary and sufficient that α > 1/p. This is, essentially, an extension of the Erdös-Turan theorem on L2 convergence. In an earlier paper, we analyzed convergence for all p > 1, showing the necessity and sufficiency of using the weighting factor 1 + Q for all p > 4. Our proofs of convergence are based on converse quadrature sum estimates, that are established using methods of H. König.

Keywords

42C1542C0565D05Erdős weightsLagrange interpolationmean convergenceLp norms
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 4 , 01 August 1996 , pp. 737 - 757
Copyright
Copyright © Canadian Mathematical Society 1996

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