A note on Tchakaloff’s Theorem

Putinar, Mihai

doi:10.1090/s0002-9939-97-03862-8

Cited by 57 publications

(46 citation statements)

References 7 publications

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“…The following Lemma, due to Blekherman and Fialkow [5], explains the key role played by discrete measures in truncated moment problems. It is a generalization of results of Tchakaloff [32] and Putinar [30]. We include a proof for the reader's benefit.…”

Section: Representability Via Discrete Measuresmentioning

confidence: 76%

Approximate super-resolution of positive measures in all dimensions

García¹,

Hernández

Junca³

et al. 2021

Applied and Computational Harmonic Analysis

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Section: Representability Via Discrete Measuresmentioning

confidence: 76%

Approximate super-resolution of positive measures in all dimensions

García¹,

Hernández

Junca³

et al. 2021

Applied and Computational Harmonic Analysis

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“…(Section 3.3) It also provides a bound on the number of nodes of minimal cubatures, namely,

((), \binom{d + n}{n})

, where d is the degree of the cubature. Analogous results were later proved with less constraint on the measure μ, and recently, this upper bound was revisited for cubatures in the plane . However, the proof of Tchakaloff's theorem is not constructive.…”

Section: Introductionmentioning

confidence: 91%

“…The above theorem was generalized in Refs. and by reducing the hypothesis on the measure μ. The theorem remains true for any finite dimensional spaces of polynomials.…”

Section: Algorithms To Determine Cubaturesmentioning

confidence: 99%

Algorithms for computing cubatures based on moment theory

Collowald

Hubert

2018

Stud Appl Math

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Quadrature is an approximation of the definite integral of a function by a weighted sum of function values at specified points, or nodes, within the domain of integration. Gaussian quadratures are constructed to yield exact results for any polynomials of degree 2r−1 or less by a suitable choice of r nodes and weights. Cubature is a generalization of quadrature in higher dimension. In this article, we elaborate algorithms to compute all minimal cubatures for a given domain and a given degree. We propose first algorithm in symbolic computation to characterize all cubatures of a given degree with a fixed number of nodes. The determination of the nodes and weights is then left to the computation of the eigenvectors of the matrix identified at the characterization stage and can be performed numerically. The characterization of cubatures on which our algorithms are based stems from moment theory. We formulate the results there in a basis‐independent way: Rather than considering the moment matrix, the central object in moment problems, we introduce the underlying linear map from the polynomial ring to its dual, the Hankel operator. This makes natural the use of bases of polynomials other than the monomial basis, and proves to be computationally relevant, either for numerical properties or to exploit symmetry.

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“…can be compressed into a formula with the same properties, whose nodes are a re-weighted subset of the original ones with cardinality not exceeding N . We focus here on the bivariate discrete case, recalling however that this result is valid in full generality for any discrete or continuous measure in any dimension, by the celebrated Tchakaloff's theorem; cf., e.g., [33]. We give below the main lines of the construction, following [31] where the general case of discrete measures in any dimension is treated.…”

Section: Caratheodory-tchakaloff Cubature Compressionmentioning

confidence: 99%

Algebraic cubature on polygonal elements with a circular edge

Artioli

Sommariva

Vianello

2020

Computers & Mathematics with Applications

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We ccompute low-cardinality algebraic cubature formulas on convex or concave polygonal elements with a circular edge, by subdivision into circular quadrangles, blending formulas via subperiodic trigonometric Gaussian quadrature and final compression via Caratheodory-Tchakaloff subsampling of discrete measures. We also discuss applications to the VEM (Virtual Element Method) in computational mechanics problems.

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A note on Tchakaloff’s Theorem

Abstract: Abstract. A classical result of Tchakaloff on the existence of exact quadrature formulae up to a given degree is extended to positive measures without compact support. A criterion for the existence of Gaussian quadratures for a class of such measures is also derived from the main proof.

Cited by 57 publications

References 7 publications

Approximate super-resolution of positive measures in all dimensions

Approximate super-resolution of positive measures in all dimensions

Algorithms for computing cubatures based on moment theory

Algebraic cubature on polygonal elements with a circular edge

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