Can a minimal degree 6 cubature rule for the disk have all points inside?

Easwaran, C. V.; Fialkow, Lawrence A.; Petrović, Srdjan

doi:10.1016/j.cam.2005.02.001

Cited by 3 publications

(2 citation statements)

References 20 publications

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“…Minimal representing measures supported on prescribed sets arise when one wants to construct a minimal cubature formula. For a moment matrix approach see [12], [11]. For a different approach see [17].…”

Section: Introductionmentioning

confidence: 99%

The truncated matrix-valued $K$-moment problem on $\mathbb {R}^d$, $\mathbb {C}^d$, and $\mathbb {T}^d$

Kimsey

Woerdeman

2013

Trans. Amer. Math. Soc.

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The truncated matrix-valued K-moment problem on R d , C d , and T d will be considered. The truncated matrix-valued K-moment problem on R d requires necessary and sufficient conditions for a multisequence of Hermitian matrices {S γ } γ∈Γ (where Γ is a finite subset of N d 0) to be the corresponding moments of a positive Hermitian matrix-valued Borel measure σ, and also the support of σ must be contained in some given non-empty set K ⊆ R d , i.e., (0.1) S γ = R d ξ γ dσ(ξ), for all γ ∈ Γ, and (0.2) supp σ ⊆ K. Given a non-empty set K ⊆ R d and a finite multisequence, indexed by a certain family of finite subsets of N d 0 , of Hermitian matrices we obtain necessary and sufficient conditions for the existence of a minimal finitely atomic measure which satisfies (0.1) and (0.2). In particular, our result can handle the case when Γ = {γ ∈ N d 0 : 0 ≤ |γ| ≤ 2n + 1}. We will also discuss a similar result in the multivariable complex and polytorus setting.

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Section: Introductionmentioning

confidence: 99%

The truncated matrix-valued $K$-moment problem on $\mathbb {R}^d$, $\mathbb {C}^d$, and $\mathbb {T}^d$

Kimsey

Woerdeman

2013

Trans. Amer. Math. Soc.

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“…• The bad reason is that our proofs will in general not be able to guarantee the nodes to be contained in the support. • There are examples where the minimal number of nodes can only be reached if one does not insist on all nodes being contained in the support, e.g., when one searches a quadrature rule of degree 6 for the Lebesgue measure on the closed unit disk [EFP,Theorem 1.1]. • For many important measures (e.g., the Gaussian measure on R n ) the requirement to lie in the support is anyway empty and it seems more important that the nodes are not too "far out", at least those with significant weight.…”

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confidence: 99%

Optimization approaches to quadrature: New characterizations of Gaussian quadrature on the line and quadrature with few nodes on plane algebraic curves, on the plane and in higher dimensions

Riener

Schweighofer

2018

Journal of Complexity

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Let d and k be positive integers. Let µ be a positive Borel measure on R 2 possessing finite moments up to degree 2d − 1. If the support of µ is contained in an algebraic curve of degree k, then we show that there exists a quadrature rule for µ with at most dk many nodes all placed on the curve (and positive weights) that is exact on all polynomials of degree at most 2d − 1. This generalizes both Gauss and (the odd degree case of) Szegő quadrature where the curve is a line and a circle, respectively, to arbitrary plane algebraic curves. We use this result to show that, without any hypothesis on the support of µ, there is always a cubature rule for µ with at most 3 2 d(d − 1) + 1 many nodes. In both results, we show that the quadrature or cubature rule can be chosen such that its value on a certain positive definite form of degree 2d is minimized. We characterize the unique Gaussian quadrature rule on the line as the one that minimizes this value or several other values as for example the sum of the nodes' distances to the origin. The tools we develop should prove useful for obtaining similar results in higher-dimensional cases although at the present stage we can present only partial results in that direction. CONTENTS

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A Moment Matrix Approach to Multivariable Cubature

Fialkow

Petrović

2005

Integr. equ. oper. theory

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Abstract. We develop an approach to multivariable cubature based on positivity, extension, and completion properties of moment matrices. We obtain a matrix-based lower bound on the size of a cubature rule of degree 2n + 1; for a planar measure µ, the bound is based on estimating ρ(C) := inf{rank (T −C) : T Toeplitz and T ≥ C}, where C := C [µ] is a positive matrix naturally associated with the moments of µ. We use this estimate to construct various minimal or near-minimal cubature rules for planar measures. In the case when C = diag (c1, . . . , cn) (including the case when µ is planar measure on the unit disk), ρ(C) is at least as large as the number of gaps c k > c k+1 .

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Can a minimal degree 6 cubature rule for the disk have all points inside?

Cited by 3 publications

References 20 publications

The truncated matrix-valued $K$-moment problem on $\mathbb {R}^d$, $\mathbb {C}^d$, and $\mathbb {T}^d$

The truncated matrix-valued $K$-moment problem on $\mathbb {R}^d$, $\mathbb {C}^d$, and $\mathbb {T}^d$

Optimization approaches to quadrature: New characterizations of Gaussian quadrature on the line and quadrature with few nodes on plane algebraic curves, on the plane and in higher dimensions

A Moment Matrix Approach to Multivariable Cubature

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