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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| Published in: | American journal of mathematics Vol. 140; no. 3; pp. 789 - 820 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Baltimore
Johns Hopkins University Press
01.06.2018
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| ISSN: | 0002-9327, 1080-6377, 1080-6377 |
| Online Access: | Get full text |
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| 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 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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| AbstractList | 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. 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. 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. © 2018 by Johns Hopkins University Press. |
| Author | Berman, Robert J. Ortega-Cerdà, Joaquim |
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| BackLink | https://gup.ub.gu.se/publication/269919$$DView record from Swedish Publication Index (Göteborgs universitet) https://research.chalmers.se/publication/503495$$DView record from Swedish Publication Index (Chalmers tekniska högskola) |
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| ContentType | Journal Article |
| Contributor | Universitat de Barcelona |
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| Copyright | 2018 by Johns Hopkins University Press Copyright © The Johns Hopkins University Press. Copyright Johns Hopkins University Press Jun 2018 (c) Johns Hopkins University Press, 2018 info:eu-repo/semantics/openAccess |
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| Snippet | 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... |
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| SubjectTerms | 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 |
| Title | SAMPLING OF REAL MULTIVARIATE POLYNOMIALS AND PLURIPOTENTIAL THEORY |
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