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Optimal approximants and orthogonal polynomials in several variables

Part of: Linear function spaces and their duals Approximations and expansions

Published online by Cambridge University Press:  26 November 2020

Meredith Sargent  and
Alan A. Sola
Meredith Sargent
Affiliation:
Department of Mathematics, University of Arkansas, Fayetteville, AR, USA e-mail: sargent@uark.edu
Alan A. Sola*
Affiliation:
Department of Mathematics, Stockholm University, Stockholm, Sweden
*
e-mail: sola@math.su.se
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Abstract

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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.

Keywords

Optimal approximantsorthogonal polynomialsweakly inner functionszero sets

MSC classification

Primary: 41A10: Approximation by polynomials
Secondary: 46E22: Hilbert spaces with reproducing kernels (=
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 2 , April 2022 , pp. 428 - 456
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Canadian Mathematical Society 2020

Footnotes

A.A.S acknowledges support from Ivar Bendixons stipendiefond för docenter.

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