Bounds for polynomials with a unit discrete norm

Abstract

Let E be the set of N equidistant points in (-1,1) and ℙ n(E) be the set of all polynomials P of degree ≤ n with max{|P(ζ)|, ζ ∈ E} ≤ 1. We prove that Kn,N(x) = maxP∈ℙn(E)|P(x) ≤ C log φ/arctan (N/n √r 2 - x2), |x| ≤ r := √1 - n2/N 2 where n < N and C is an absolute constant. The result is essentially sharp. Bounds for Kn,N(z), z ε ℂ, uniform for n < N, are also obtained. The method of proof of those results is a general one. It allows one to obtain sharp, or sharp up to a log N factor, bounds for Kn,N under rather general assumptions on E (#E = N). A "model" result is announced for a class of sets E. Main components of the method are discussed in some detail in the process of investigating the case of equally spaced points.