Optimal Polynomial Prediction Measures and Extremal Polynomial Growth

TL;DR

This paper establishes a connection between optimal polynomial prediction measures and extremal polynomial growth, providing explicit solutions for specific cases like the interval [-1,1] at imaginary points.

Contribution

It introduces a novel equivalence between minimal variance measures and extremal polynomial growth problems, extending classical results to complex points.

Findings

Derived extremal polynomials for the interval [-1,1] at imaginary points

Linked polynomial extremal growth to minimal variance prediction measures

Extended classical extremal polynomial results to complex external points

Abstract

We show that the problem of finding the measure supported on a compact subset K of the complex plane such that the variance of the least squares predictor by polynomials of degree at most n at a point exterior to 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 this external point. We use this to find the polynomials of extremal growth for the interval [-1,1] at a purely imaginary point. The related problem on the extremal growth of real polynomials was studied by Erd\H{o}s in 1947.