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

Published online by Cambridge University Press:  20 November 2018

S. B. Damelin  and
D. S. Lubinsky
S. B. Damelin
Affiliation:
Department of Mathematics University of the Witwatersrand P.O. Wits 2050 Republic of South Africa
D. S. Lubinsky
Affiliation:
Department of Mathematics University of the Witwatersrand P.O. Wits 2050 Republic of South Africa
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Abstract

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We investigate mean convergence of Lagrange interpolation at the zeros of orthogonal polynomials pn(W2, x) for Erdös 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 < ∞, Δ ∊ ℝ, k > 0. 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

Keywords

42C1542C0565D05Erdős weightsLagrange interpolationmean convergenceLp norms
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 4 , 01 August 1996 , pp. 710 - 736
Copyright
Copyright © Canadian Mathematical Society 1996

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