Optimal approximants and orthogonal polynomials in several variables

We discuss the notion of optimal polynomial approximants in multivariable reproducing kernel Hilbert spaces. In particular, we analyze difficulties that arise in the multivariable case which are not present in one variable, for example, a more complicated relationship between optimal approximants an...

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Bibliographic Details
Published in:arXiv.org 2020-03
Main Authors: Sargent, Meredith, Sola, Alan
Format: Article
Language:English
Subjects:
Approximants
Functions (mathematics)
Hilbert space
Mathematical analysis
Polynomials
Online Access:Get full text
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Summary:We discuss the notion of optimal polynomial approximants in multivariable reproducing kernel Hilbert spaces. In particular, we analyze difficulties that arise in the multivariable case which are not present in one variable, for example, a more complicated relationship between optimal approximants and orthogonal polynomials in weighted spaces. Weakly inner functions, whose optimal approximants are all constant, provide extreme cases where nontrivial orthogonal polynomials cannot be recovered from the optimal approximants. Concrete examples are presented to illustrate the general theory and are used to disprove certain natural conjectures regarding zeros of optimal approximants in several variables.
ISSN:2331-8422
DOI:10.48550/arxiv.2002.08790