Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 842.41003
Autor: Erdös, Paul; Szabados, J.; Vértesi, P.
Title: On the integral of the Lebesgue function of interpolation. II. (In English)
Source: Acta Math. Hung. 68, No.1-2, 1-6 (1995).
Review: Notations. -1 \leq x0,n < x1,n < ... < x1,n < 1 is a set of nodes on the interval [-1,1]. For brevity set xk: = xk,n. Define also some well known quantities \ellk(x): = \ellk,n(x) = {{\omega(x)} \over {\omega' (xk) (x- xk)}}, \omega (x) = prodnk = 0 (x-xk), \lambda(a,b): = maxa \leq x \leq b sumnk = 1 |\ellk (x)|, -1 \leq a < b \leq 1. The present paper and a former paper by the first two authors [Acta Math. Acad. Sci. Hungar 32, 191-195 (1978; Zbl 391.41003)] deal with lower bound estimates of the function \lambdan(x): = sumnk = 0 |\ellk(x)|. In the above mentioned paper it was shown that for any interval [a,b]\subseteq [-1,1] and arbitrary nodes xk the inequality
intab sumnk = 0 |\ellk(x)| dx \geq c(b- a) log n holds for sufficiently large n depending only on the interval [a,b ]. This inequality was an improvement of Bernstein's \lambdan (a,b) \geq c1 log n, n \geq n1 (a,b). A further improvement is shown in the present paper, namely a similar inequality is derived for every individual interval [an, bn ]\subseteq [-1,1] and for all n without exception. The result states
Theorem. There exists an absolute positive constant c for which the inequality
intbnan \lambda(x) dx \geq c(bn - an) log (n(\alphan - \betan)+2), (an\cos \alphan, bn = \cos \betan). In fact the authors show the sharpness, in a sense, of their estimate by showing that
maxan \leq x \leq bn \lambdan (x) = O(log(n (\alphan - \betan)+2)).
Reviewer: Z.Rubinstein (Haifa)
Classif.: * 41A05 Interpolation
Citations: Zbl 391.41003
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