Optimal Polynomial Prediction Measures and Extremal Polynomial Growth

Abstract

We show that the problem of finding the measure supported on a compact set $K\subset \C$ such that the variance of the least squares predictor by polynomials of degree at most $n$ at a point $z_0\in\C^d\backslash K$ is a minimum, is equivalent to the problem of finding the polynomial of degree at most $n,$ bounded by 1 on $K,$ with extremal growth at $z_0.$ We use this to find the polynomials of extremal growth for $[-1,1]\subset \C$ at a purely imaginary point. The related problem on the extremal growth of real polynomials was studied by Erd\H{o}s (Bull Am Math Soc 53:1169-1176, 1947).