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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