Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 499.41004
Autor: Erdös, Pál; Vertesi, Peter
Title: On the Lebesgue function of interpolation. (In English)
Source: Proc. 13th Symp. Ring theory, Okayama 1980, 299-309 (1981).
Review: [For the entire collection see Zbl 436.00014.]
One consider a triangular matrix Z = {xk,n} (n in \Cal{N},k = 1(1)n) of distinct real arbitrary nodes, such that: 1 \leq xn,n < xn-i,n < ... < x1,n \leq 1. Let \ellk(x) = \ellk,n(Z,x) to be corresponding fundamental polynomials of the Lagrange interpolation. It is known that the Lebesgue function and the Lebesgue constant, defined respectively by \lambdan(x) = \lambdan(Z,x) = sumk = 1n|\ellk(x)|, \lambdan = \lambdan(Z) = max\lambdan(x) for -1 \leq x \leq 1, play a decisive role in the convergence and divergence properties of Lagrange interpolation. In 1961 {
P.Erdös} [Acta Math. Acad. Sci. Hung. 12, 235-244 (1961; Zbl 098.04103)] has proved that for any system of nodes xk,n (k = 1(1)n) we have \lambdan > 2\pi-1\ell n n-c (n \geq n0), where c is a certain positive absolute constant. In this paper the authors prove the following remarkable theorem: If \epsilon is any given positive number, then for arbitrary matrix Z there exist sets Hn, with |Hn| \leq \epsilon and \eta(\epsilon) > 0, such that \lambdan(x) > \eta(\epsilon)\ell n n, whenever x in [-1,1]|Hn and n \geq n0(\epsilon). The case of Chebyshev nodes showsthat this order is best possible. In the proof of this theorem the authors use some results from their recent common paper [ibid, 36, 71-89 (1980; Zbl 463.41002)]. Finally we mention the following important corollary of this theorem: Let \epsilon > 0 and \eta(\epsilon) > 0 be as above. If Sn\subseteq[-1,1] are arbitrary measurable sets then for any matrix Z we have intSn\lambdan(x)dx > (|Sn|-\epsilon)\eta(\epsilon)\ell n n, whenever n \geq n0(\epsilon). The special case Sn = S = [a,b] has been investigated earlier by P.Erdös and J.Szabados [ibid. 32, 191-195 (1978; Zbl 391.41003)].
Reviewer: D.D.Stancu. \end
Classif.: * 41A05 Interpolation
41A17 Inequalities in approximation
65D05 Interpolation (numerical methods)
Citations: Zbl.436.00014; Zbl.098.041; Zbl.463.41002; Zbl.391.41003
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