Weighted Lagrange and Hermite–Fejér interpolation on the real line

Abstract

For a wide class of weights, a systematic investigation of the convergence-divergence behavior of Lagrange interpolation is initiated. A system of nodes with optimal Lebesgue constant is found, and for Hermite weights an exact lower estimate of the norm of projection operators is given. In the same spirit, the case of Hermite-Fej6r interpolation is also considered.

Approximation of the Hilbert Transform on the real line using Hermite zeros

“…hold uniformly with respect to m and k. The previous relations can be easily deduced from [17] and [39]. The following lemma is new and will be useful in the sequel.…”

“…holds, where C is a positive constant independent of m and f. Theorem 2.2 is a simple but useful modification of a previous result by J. Szabados [39]. In Section 4 we will give the proof of this theorem not only for completeness, but also because we can deduce a slightly more general estimate.…”

“…We have only to prove that the sum on the right-hand side of (4.7) is dominated by log m. But it has been proved in [39].…”

“…Using an idea by J. Szabados [39], we construct a simple interpolating Lagrange polynomial and we prove that it is optimal in W ∞ 0 (see Theorem 2.2). Subsequently we approximate H(G, t) by replacing f with the above-mentioned polynomial.…”

A Lagrange-type projector on the real line

“…In order to obtain these kinds of projectors, following an idea previously used in [33,10], we denote by L m+2 (w, f ) the Lagrange polynomial interpolating f ∈ L , but onto a subspace P m+1 ⊂ P m+1 . We prove that m P m+1 is dense in L p u , for 1 ≤ p ≤ ∞, and that for each element of P m+1 , both Marcinkiewicz inequalities hold true.…”