Zentralblatt MATH

Publications of (and about) Paul Erdös
Zbl.No:  391.41003
Autor:  Erdös, Paul; Szabados, J.
Title:  On the integral of the Lebesgue function of interpolation. (In English)
Source:  Acta Math. Acad. Sci. Hung. 32, 191-195 (1978).
Review:  Let -1 \leq X1 < X2 < ... < Xn \leq 1 be n distinct numbers in (-1,+1). \omega(x) = prodi = 1n(X-Xi), put

\ellk(X) = \frac{\omega(X)}{\omega'(Xk](X-Xk)}.

\ellk(X) are the fundamental functions of Lagrange interpolation polynomials, \ellk(Xk) = 1 and \ellk(X1) = 0 for i\ne k. The author pove

sumk = 1n int-1+1 |\ellk(X)| d(X) > c log n    (1)

for a certain absolute constant c > 0. The proof is not very simple and the best value of c is not determined. It seems a reasonable guess that asymptotically (1) is a minimum if the Xi are the roots of the Chebyshev polynomial Tn(X). But we have not been able to prove this.
Classif.:  * 41A05 Interpolation
Keywords:  Lebesgue function; integration of interpolation functions; Lagrange interpolation polynomials

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