Optimal Polynomial Prediction Measures and Extremal Polynomial Growth
We show that the problem of finding the measure supported on a compact set $K\subset \C$ such that the variance of the least squares predictor by polynomials of degree at most $n$ at a point $z_0\in\C^d\backslash K$ is a minimum, is equivalent to the problem of finding the polynomial of degree at mo...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2020 |
| País: | España |
| Institución: | Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
| Repositorio: | Recercat. Dipósit de la Recerca de Catalunya |
| OAI Identifier: | oai:recercat.cat:2445/181476 |
| Acceso en línea: | https://hdl.handle.net/2445/181476 |
| Access Level: | acceso abierto |
| Palabra clave: | Desigualtats (Matemàtica) Teoria de l'aproximació Funcions de variables complexes Inequalities (Mathematics) Approximation theory Functions of complex variables |
| Sumario: | We show that the problem of finding the measure supported on a compact set $K\subset \C$ such that the variance of the least squares predictor by polynomials of degree at most $n$ at a point $z_0\in\C^d\backslash K$ 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 $[-1,1]\subset \C$ at a purely imaginary point. The related problem on the extremal growth of real polynomials was studied by Erd\H{o}s (Bull Am Math Soc 53:1169-1176, 1947). |
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