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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 491.41001

**Autor: ** Erdös, Paul; Vertesi, P.

**Title: ** On the almost everywhere divergence of Lagrange interpolation. (In English)

**Source: ** Approximation and function spaces, Proc. int. Conf., Gdansk 1979, 270-278 (1981).

**Review: ** [For the entire collection see Zbl 471.00018.]

In a previously published paper, *P.Erdös* [Acta Math. Acad. Sci. Hung. 9, 381-388 (1958; Zbl 083.29001)] stated without proof that if (x_{kn})_{1 \leq k \leq n,1 \leq n} denotes a triangular matrix of knots in the compact internal I = [-1,+1] ordered such that x_{n,n} < x_{n-1,n} < ... < x_{1,n} (n \geq 1) holds then there exists a continuous function f: I ––> **R** such that the sequence (L_{n}f)_{n \geq 1} of Lagrange interpolation polynomials L_{n} = **sum**_{1 \leq k \leq n}f(x_{kn})\ell_{kn} diverges almost everywhere in I. In fact **lim**.\sup{n ––> oo}|L_{n}f(x)| = oo for almost all x in I. The authors give a brief account of the preliminary results in this direction (G. Faber, S. Bernstein, G. Grünwald, J. Marcinkiewicz, A. A. Privalov, P. Turán, P. Erdös) and point out a sketch of the proof. The detailed proof is rather long and quite complicated, although it uses only elementary techniques. One of its important ingredients is the following result: Lemma. Let A > 0 be an arbitrary fixed number and consider an arbitrary integer m \geq m_{0}(A). For any integer n \geq n_{0}(m) exists a set H_{n}\subset I for which meas(H_{n}) \leq 1/ ln\ln m and **sum**_{\over{1 \leq k \leq n}{x_{k}\not in I_{j(x),n}}}|\ell_{kn}(x)| \geq (ln m)^{1/3}geq2A (n \geq n_{0}(m)) where I_{j(x),m} denotes that interval of the equivalent partition of I in m subintervals which contains the point x in I|H_{n}. – Details of the proof (about 300 pages) will be published in a paper to appear in Acta Math. Acad. Sci. Hung.

**Reviewer: ** W.Schempp

**Classif.: ** * 41A05 Interpolation

**Keywords: ** almost everywhere divergence; triangular matrix of knots; Lagrange interpolation polynomials

**Citations: ** Zbl.471.00018; Zbl.083.290

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