SAMPLING OF REAL MULTIVARIATE POLYNOMIALS AND PLURIPOTENTIAL THEORY

We consider the problem of stable sampling of multivariate real polynomials of large degree in a general framework where the polynomials are defined on an affine real algebraic variety 𝛭, equipped with a weighted measure. In particular, this framework contains the well-known setting of trigonometric...

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Bibliographic Details
Published in:American journal of mathematics Vol. 140; no. 3; pp. 789 - 820
Main Authors: Berman, Robert J., Ortega-Cerdà, Joaquim
Format: Journal Article
Language:English
Published: Baltimore Johns Hopkins University Press 01.06.2018
Subjects:
Algebra
Anàlisi de Fourier
Anàlisi harmònica
Density
Fourier analysis
Funcions de diverses variables complexes
Functions of several complex variables
Harmonic analysis
Matematik
Mathematical analysis
Mathematical sciences
Polynomials
Sampling
Toruses
ISSN:0002-9327, 1080-6377, 1080-6377
Online Access:Get full text
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Summary:We consider the problem of stable sampling of multivariate real polynomials of large degree in a general framework where the polynomials are defined on an affine real algebraic variety 𝛭, equipped with a weighted measure. In particular, this framework contains the well-known setting of trigonometric polynomials (when 𝛭 is a torus equipped with its invariant measure), where the limit of large degree corresponds to a high frequency limit, as well as the classical setting of one-variable orthogonal algebraic polynomials (when 𝛭 is the real line equipped with a suitable measure), where the sampling nodes can be seen as generalizations of the zeros of the corresponding orthogonal polynomials. It is shown that a necessary condition for sampling, in the general setting, is that the asymptotic density of the sampling points is greater than the density of the corresponding weighted equilibrium measure of 𝛭, as defined in pluripotential theory. This result thus generalizes the well-known Landau type results for sampling on the torus, where the corresponding critical density corresponds to the Nyqvist rate, as well as the classical result saying that the zeros of orthogonal polynomials become equidistributed with respect to the logarithmic equilibrium measure, as the degree tends to infinity.
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ISSN:0002-9327
1080-6377
1080-6377
DOI:10.1353/ajm.2018.0019