Optimal Polynomial Prediction Measures and Extremal Polynomial Growth
Classical Analysis and ODEs
2022-10-04 v1 Statistics Theory
Statistics Theory
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.
Cite
@article{arxiv.1912.12462,
title = {Optimal Polynomial Prediction Measures and Extremal Polynomial Growth},
author = {L. Bos and N. Levenberg and J. Ortega-Cerda},
journal= {arXiv preprint arXiv:1912.12462},
year = {2022}
}