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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 186.37902

**Autor: ** Erdös, Pál

**Title: ** On the boundedness and unboundedness of polynomials (In English)

**Source: ** J. Anal. Math. 19, 135-148 (1967).

**Review: ** Let x_{i}^{(j)}, 1 \leq i \leq j be numbers in the closed intervall [-1,1] strictly increasing with i for each fixed j. For each n let P_{n} denote a polynomial of degree n in x. The author proves a necessary and sufficient condition on the triangular matrix (x_{i}^{(j)}) that the following implication hold. If for each m, n(1+c) < m, and for each i, 1 \leq i \leq m, we have | P_{n} (x_{i}^{(n)})| \leq 1, then there exists a function A(c) depending only on c such that **max**(|P_{n}(x)|: -1 \leq x \leq 1) is less than A(c).

The proof is difficult, and is related with earlier work of the same author [cf. the author, Ann. of Math., II. Ser. 44, 330-337 (1943; Zbl 063.01266)]. The result proved extends results of Zygmund and Berstein concerning the Tchebycheff and Legendre polynomials respectively.

**Reviewer: ** H.J.Biesterfeldt

**Classif.: ** * 26C05 Polynomials: analytic properties (real variables)

33C25 Orthogonal polynomials and functions

**Index Words: ** approximation and series expansion

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