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Title: Sampling of real multivariate polynomials and pluripotential theory
Author: Berman, Robert J.
Ortega Cerdà, Joaquim
Keywords: Funcions de diverses variables complexes
Anàlisi harmònica
Anàlisi de Fourier
Functions of several complex variables
Harmonic analysis
Fourier analysis
Issue Date: 1-Jun-2018
Publisher: Johns Hopkins University Press
Abstract: 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 $M$, equipped with a weighted measure. In particular, this framework contains the well-known setting of trigonometric polynomials (when $M$ 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 $M$ 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 $M$, 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.
Note: Reproducció del document publicat a:
It is part of: American Journal of Mathematics, 2018, vol. 140, num. 3, p. 789-820
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ISSN: 0002-9327
Appears in Collections:Articles publicats en revistes (Matemàtiques i Informàtica)

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