Remark on a Theorem about Polynomials

Abstract

1. Let

$$ {t_{mm}} < {t_{m - 1,m}} < \cdots < {t_{0m}},{\text{ }}m = 1,2, \ldots $$

(1)

be equidistant nodes on the real line. Consider the set P _n(m) of real polynomials P(x) of degree at most n with $\left| {P\left( {{t_{km}}} \right)} \right| \leqslant 1,0 \leqslant k \leqslant m.$. Let

$$ \left\| P \right\| = \mathop {\max }\limits_{{t_{mm \leqslant x \leqslant {t_{{0_m}}}}}} \left| {P\left( x \right)} \right| $$

(2)

$$ b\left( {n,m} \right) = \mathop {\sup }\limits_{P \in {\mathcal{P}_n}\left( m \right)} \left\| P \right\|. $$

(3)

Supported by the Hungarian National Science Foundation Grants Nos. 1910 and T7570.