Optimal Polynomial Prediction Measures and Extremal Polynomial Growth

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 mos...

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Bibliographic Details
Published inarXiv.org
Main Authors Bos, L, Levenberg, N, Ortega-Cerda, J
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 28.12.2019
Subjects
Mathematics - Classical Analysis and ODEs
Mathematics - Statistics Theory
Polynomials
Statistics - Theory
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Summary: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 Erdős in 1947.
ISSN:2331-8422
DOI:10.48550/arxiv.1912.12462