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 (xkn)1 \leq k \leq n,1 \leq n denotes a triangular matrix of knots in the compact internal I = [-1,+1] ordered such that xn,n < xn-1,n < ... < x1,n  (n \geq 1) holds then there exists a continuous function f: I ––> R such that the sequence (Lnf)n \geq 1 of Lagrange interpolation polynomials Ln = sum1 \leq k \leq nf(xkn)\ellkn diverges almost everywhere in I. In fact lim.\sup{n ––> oo}|Lnf(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 m0(A). For any integer n \geq n0(m) exists a set Hn\subset I for which meas(Hn) \leq 1/ ln\ln m and

sum\over{1 \leq k \leq n{xk\not in Ij(x),n}}|\ellkn(x)| \geq (ln m)1/3geq2A  (n \geq n0(m))

where Ij(x),m denotes that interval of the equivalent partition of I in m subintervals which contains the point x in I|Hn. – 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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