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On the Lebesgue function of weighted Lagrange interpolation. I: Freud-type weights. (English) Zbl 0945.41003

In this interesting paper, the author investigates the Lebesgue functions of interpolation for Freud weights and general interpolation arrays. The basic (and important) conclusion is that most of the time, the Lebesgue function associated with the \(n\)th row of an array of interpolation points has to grow at least as fast as \(\log n\). To make this more precise, let \(X\) be an array of interpolation points on the real line, so that the \(n\)th row of \(X\) has the form \[ -\infty <x_{nn} <x_{n-1,n}< \cdots <x_{1n} < \infty. \] Let \[ w(x): =\exp\bigl( -Q(x)\bigr), \quad x\in\mathbb{R}, \] be a Freud weight. This means that \(Q\) is even, of “smooth” polynomial growth at \(\infty \), amongst other things. The most important examples are \[ w(x)= \exp\bigl(- |x|^\alpha \bigr), \quad \alpha>1. \] Let \(\{\ell_{jn} \}^n_{j=1}\) be the fundamental polynomials of Lagrange interpolation at \(\{x_{jn} \}^n_{j =1}\), so that \[ \ell_{jn} (x_{kn})= \delta_{jk}, \quad 1\leq j,\;k\geq n. \] Then the Lagrange interpolating polynomial to a function \(f:\mathbb{R} \to\mathbb{R}\) at \(\{x_{ jn}\}^n_{j=1}\) is \[ L_n[f](x)= \sum^n_{j=1} f(x_{jn}) \ell_{jn} (x). \] The norm of this linear functional on a suitable weighted space is the Lebesgue function \[ \lambda_n (W,X,x):=W(x) \sum^n_{k=1} \bigl|\ell_{kn}(w) \bigr|W^{-1} (x_{kn}). \] The author shows that for a large class of weights \(W\), any \(\varepsilon>0\), and any array \(X\), we have, at least for large enough \(n\), \[ \lambda_n(W,X,x)\geq{\varepsilon\over 3840}\log n,\;x\in [-a_n,a_n] \setminus {\mathcal E}_n \] where \([-a_n,a_n]\) is the Mhaskar-Rakhmanov-Saff interval for \(W\) and the linear Lebesgue measure of \({\mathcal E}_n\) is at most \(2a_n\varepsilon\).

MSC:

41A05 Interpolation in approximation theory
41A10 Approximation by polynomials
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