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

Damelin, S. B. ; Lubinsky, D. S.

Canadian journal of mathematics, Tome 48 (1996) no. 4, pp. 737-757

Résumé

We complete our investigations of mean convergence of Lagrange interpolation at the zeros of orthogonal polynomials pn (W 2, x) for Erdős weights W 2 = 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 (W 2, x) = pn (e -2Q , x). Then for to hold for every continuous functionƒ:R. —> R satisfying it is necessary and sufficient that α > 1/p. This is, essentially, an extension of the Erdös-Turan theorem on L 2 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.

Détail

DOI : 10.4153/CJM-1996-038-9

Mots-clés : 42C15, 42C05, 65D05, Erdős weights, Lagrange interpolation, mean convergence, Lp norms

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