Optimal Polynomial Prediction Measures and Extremal Polynomial Growth

Abstract

AbstractWe show that the problem of finding the measure supported on a compact set $$K\subset \mathbb {C}$$ K ⊂ C such that the variance of the least squares predictor by polynomials of degree at most n at a point $$z_0\in \mathbb {C}^d\backslash K$$ z 0 ∈ C d \ 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 $$z_0.$$ z 0 . We use this to find the polynomials of extremal growth for $$[-1,1]\subset \mathbb {C}$$ [ - 1 , 1 ] ⊂ 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).