Optimal Polynomial Prediction Measures and Extremal Polynomial Growth

Abstract

We show that the problem of finding the measure supported on a compact set Ksubset mathbb {C} such that the variance of the least squares predictor by polynomials of degree at most n at a point z_0in mathbb {C}^dbackslash 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 mathbb {C} at a purely imaginary point. The related problem on the extremal growth of real polynomials was studied by Erdős (Bull Am Math Soc 53:1169–1176, 1947).