Publications of (and about) Paul Erdös
Autor: Erdös, Pál
Title: On the boundedness and unboundedness of polynomials (In English)
Source: J. Anal. Math. 19, 135-148 (1967).
Review: Let xi(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 Pn denote a polynomial of degree n in x. The author proves a necessary and sufficient condition on the triangular matrix (xi(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 | Pn (xi(n))| \leq 1, then there exists a function A(c) depending only on c such that max(|Pn(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.
Classif.: * 26C05 Polynomials: analytic properties (real variables)
33C25 Orthogonal polynomials and functions
Index Words: approximation and series expansion
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