Multivariate Gaussian cubature formulae

Berens, Hubert; Schmid, Hans; Xu, Yuan

doi:10.1007/bf01193547

Cited by 42 publications

(52 citation statements)

References 9 publications

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“…A simple observation shows that, for a compactly supported positive measure in R n , the number of nodes N of a degree d quadrature formula with positive coefficients satisfies N ≥ N [d/2] (n). By a widely accepted convention and terminology, the case N = N [d/2] corresponds to Gaussian quadratures; see for details [1], [7] and [8] and the references there.…”

Section: Gaussian Quadraturesmentioning

confidence: 99%

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A note on Tchakaloff’s Theorem

Putinar

1997

Proc. Amer. Math. Soc.

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Section: Gaussian Quadraturesmentioning

confidence: 99%

“…For the rather intricate structure of the Gaussian quadrature formulae and the present status of their theory the reader can profitably consult references [1], [4], [7] and [8].…”

Section: Propositionmentioning

confidence: 99%

A note on Tchakaloff’s Theorem

Putinar

1997

Proc. Amer. Math. Soc.

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“…Similarly other sets of orthogonal polynomials can be constructed (see [37,38] and [39]). Let p m (x), m = 0, 1, 2, .…”

Section: Symmetric and Antisymmetric Sets Of Polynomialsmentioning

confidence: 99%

“…For m, m ∈D + M , the discrete functions (11.37) satisfy the orthogonality relation Proof . Due to the orthogonality relation for the cosine functions φ m (s) = 2 cos(πms) (see formula (11.28)) we have 39) where (m w(1) , m w(2) , . .…”

Section: Discrete Sine and Cosine Transformsmentioning

confidence: 99%

Antisymmetric Orbit Functions

Klimyk¹

2007

SIGMA

148

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Abstract. In the paper, properties of antisymmetric orbit functions are reviewed and further developed. Antisymmetric orbit functions on the Euclidean space E n are antisymmetrized exponential functions. Antisymmetrization is fulfilled by a Weyl group, corresponding to a Coxeter-Dynkin diagram. Properties of such functions are described. These functions are closely related to irreducible characters of a compact semisimple Lie group G of rank n. Up to a sign, values of antisymmetric orbit functions are repeated on copies of the fundamental domain F of the affine Weyl group (determined by the initial Weyl group) in the entire Euclidean space E n . Antisymmetric orbit functions are solutions of the corresponding Laplace equation in E n , vanishing on the boundary of the fundamental domain F . Antisymmetric orbit functions determine a so-called antisymmetrized Fourier transform which is closely related to expansions of central functions in characters of irreducible representations of the group G. They also determine a transform on a finite set of points of F (the discrete antisymmetric orbit function transform). Symmetric and antisymmetric multivariate exponential, sine and cosine discrete transforms are given.

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“…The numbers N min ( , d, ) and corresponding cubature formulas are only known in exceptional cases, see, e.g., Schmid (1983), Berens, Schmid and Xu (1995), and Cools (1997). Thus one is interested in upper and lower bounds for this quantity.…”

Section: Introductionmentioning

confidence: 99%

Cubature formulas for symmetric measures in higher dimensions with few points

Hinrichs

Novak

2007

Math. Comp.

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Abstract. We study cubature formulas for d-dimensional integrals with an arbitrary symmetric weight function of product form. We present a construction that yields a high polynomial exactness: for fixed degree = 5 or = 7 and large dimension d the number of knots is only slightly larger than the lower bound of Möller and much smaller compared to the known constructions.We also show, for any odd degree = 2k + 1, that the minimal number of points is almost independent of the weight function. This is also true for the integration over the (Euclidean) sphere.

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Multivariate Gaussian cubature formulae

Cited by 42 publications

References 9 publications

A note on Tchakaloff’s Theorem

A note on Tchakaloff’s Theorem

Antisymmetric Orbit Functions

Cubature formulas for symmetric measures in higher dimensions with few points

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