##
**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 x_{0,n} < x_{1,n} < ... < x_{1,n} < 1 is a set of nodes on the interval [-1,1]. For brevity set x_{k}: = x_{k,n}. Define also some well known quantities \ell_{k}(x): = \ell_{k,n}(x) = {{\omega(x)} \over {\omega' (x_{k}) (x- x_{k})}}, \omega (x) = **prod**^{n}_{k = 0} (x-x_{k}), \lambda(a,b): = **max**_{a \leq x \leq b} **sum**^{n}_{k = 1} |\ell_{k} (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 \lambda_{n}(x): = **sum**^{n}_{k = 0} |\ell_{k}(x)|. In the above mentioned paper it was shown that for any interval [a,b]\subseteq [-1,1] and arbitrary nodes x_{k} the inequality

**int**_{a}^{b} **sum**^{n}_{k = 0} |\ell_{k}(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 \lambda_{n} (a,b) \geq c_{1} log n, n \geq n_{1} (a,b). A further improvement is shown in the present paper, namely a similar inequality is derived for every individual interval [a_{n}, b_{n} ]\subseteq [-1,1] and for all n without exception. The result states

Theorem. There exists an absolute positive constant c for which the inequality

**int**^{bn}_{an} \lambda(x) dx \geq c(b_{n} - a_{n}) log (n(\alpha_{n} - \beta_{n})+2), (a_{n}\cos \alpha_{n}, b_{n} = \cos \beta_{n}). In fact the authors show the sharpness, in a sense, of their estimate by showing that

**max**_{an \leq x \leq bn} \lambda_{n} (x) = O(log(n (\alpha_{n} - \beta_{n})+2)).

**Reviewer: ** Z.Rubinstein (Haifa)

**Classif.: ** * 41A05 Interpolation

**Citations: ** Zbl 391.41003

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag