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**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 = **{**x_{k,n}**}** (n in \Cal{N},k = 1(1)n) of distinct real arbitrary nodes, such that: 1 \leq x_{n,n} < x_{n-i,n} < ... < x_{1,n} \leq 1. Let \ell_{k}(x) = \ell_{k,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 \lambda_{n}(x) = \lambda_{n}(Z,x) = **sum**_{k = 1}^{n}|\ell_{k}(x)|, \lambda_{n} = \lambda_{n}(Z) = **max**\lambda_{n}(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 x_{k,n} (k = 1(1)n) we have \lambda_{n} > 2\pi^{-1}\ell n n-c (n \geq n_{0}), 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 H_{n}, with |H_{n}| \leq \epsilon and \eta(\epsilon) > 0, such that \lambda_{n}(x) > \eta(\epsilon)\ell n n, whenever x in [-1,1]|H_{n} and n \geq n_{0}(\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 S_{n}\subseteq[-1,1] are arbitrary measurable sets then for any matrix Z we have **int**_{Sn}\lambda_{n}(x)dx > (|S_{n}|-\epsilon)\eta(\epsilon)\ell n n, whenever n \geq n_{0}(\epsilon). The special case S_{n} = 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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