Polynomials small at N equidistant points in (-1,1) are small on subinterval

You need to know: Polynomial, degree of a polynomial, inverse tangent function $\arctan(x)$ .

Background: Let $P_{n,N}$ be the set of polynomials P of degree at most n with $|P(x_k)|\leq 1, k=1,2,\dots,N$ , where $x_k = -1 + (2k-1)/N, \, k=1,2,\dots,N$ is the sequence of N equidistant points on $(-1,1)$ . Let $K_{n,N}(x) = \max\limits_{P\in P_{n,N}}|P(x)|$ .

The Theorem: On 22nd November 2002, Evguenii Rakhmanov submitted to the Annals of Mathematics a paper in which he proved the existence of constant C such that inequality $K_{N,n}(x) \leq C \log \frac{\pi}{\arctan \left(\frac{N}{n}\sqrt{r^2-x^2}\right)}$ holds for all $n<N$ and all $x\in(-r,r)$ , where $r=\sqrt{1-\frac{n^2}{N^2}}$ .

Short context: The Theorem implies that if a degree n polynomial is bounded at N pre-specified discrete points then its maximal possible absolute value $K_{N,n}(x)$ is bounded on any compact subinterval of $(-r,r)$ . The bound is essentially sharp. It is useful in approximation theory, where we approximate a function by a polynomial at some discrete points and need to prove that this approximation works well on some interval.