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 inCanadian journal of mathematics Vol. 74; no. 2; pp. 428 - 456
Main Authors Sargent, Meredith, Sola, Alan A.
Format Journal Article
LanguageEnglish
Published Canada Canadian Mathematical Society 01.04.2022
Cambridge University Press
Subjects
Functions (mathematics)
Hilbert space
Mathematical analysis
Optimal approximants
orthogonal polynomials
Polynomials
weakly inner functions
zero sets
weakly inner functions
Optimal approximants
46E22
zero sets
41A10
orthogonal polynomials
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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:0008-414X
1496-4279
1496-4279
DOI:10.4153/S0008414X20000826