Results 121 to 130 of about 173 (155)

Cubature Formulae, Polytopes, and Spherical Designs

1981
The construction of a cubature formula of strength t for the unit sphere Ω d in ℝ d amounts to finding finite sets X 1,..., X N ⊂ Ω d and coefficients a 1,..., a N ∈ ℝ such that|Ωd|−1∫Ωdf(ξ)dω(ξ)=∑i=1Nai|Xi|−1∑x∈Xιf(x),(1.1)for all functions f represented on Ω d by polynomials of degree ⩽ t; cf. [16], [15], [11]. Sobolev [14,15] introduced group theory
Goethals, J.M., Seidel, J.J.
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Multivariate Gaussian cubature formulae

Archiv der Mathematik, 1995
Gaussian quadrature can be briefly described as follows: There is a one- parameter family of minimal formulae of degree \(2n-2\) the nodes of which are the roots of \(p_ n+ \rho p_{n-1}\), \(\rho\neq 0\), where \(\{p_ n\}_{n=0}^ \infty\), denotes the system of orthogonal polynomials w.r.t.
Berens, H., Schmid, H. J., Xu, Y.
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Theory of Cubature Formulas

2006
The problems of the theory of cubature formulas, when we study their error functionals in the corresponding functional spaces, can be treated as problems of functional analysis. In particular, applying certain Hilbert metrics and solving the appearing problems by the methods of variations calculus, we can obtain exact estimates of the norms of error ...
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Fast computation of cubature formulae for the sphere

2017 Hands-free Speech Communications and Microphone Arrays (HSCMA), 2017
The near-uniform distribution of nodes on the surface of a sphere has found many uses in numerical integration, physics, chemistry, crystallography, and more recently in the capture, representation and reproduction of spatial audio. A popular solution posed by Fliege and Meyer treats nodes as charged particles that are constrained to lie on the surface
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On cubature formulas on the basis of lattices

Analysis Mathematica, 1995
Construct a sequence of lattice rules for the upper estimation of the value \[ M( P(S_\gamma (n)))= \sup\{|t(\cdot) |_\infty/ |t(\cdot) |_1: t\in P(S_\gamma (n)),\;t\not\equiv 0\} \] for \(n=2\). \(P(S_\gamma (n))\) denotes the class of all nonnegative trigonometric polynomials \(t(x)\) with the spherical spectrum \(S_\gamma(n)\).
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Cubature Formulas of Finite Order

1997
We speak about a cubature formula of infinite order whenever the error of the formula is O(h m ) for all integer m and all functions in the space under study. Here h is the mesh-size of the lattice of integration. The Mean Value Theorem for harmonic functions provides the simplest example of a formula of such kind. In § 6 and § 8 of the current section
S. L. Sobolev, V. L. Vaskevich
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Cubature formulae for Besov classes

Izvestiya: Mathematics, 1997
Summary: This paper is devoted to an investigation of optimal cubature formulae for Besov classes of functions with restrictions on the mixed difference. This study is a continuation of papers [Mat. Zametki 49, No. 1, 149-151 (1991; Zbl 0760.41017)] and Mat. Sb. 183, No. 7, 23-34 (1992; Zbl 0786.41025)].
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Cubature Formulas with Regular Boundary Layer

2006
In this chapter we show that cubature formulas with regular boundary layer are asymptoticallymath-optimal as the lattice mesh-size vanishes.
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Cubature Formulae for Polyharmonic Functions

2001
We construct cubature formulae for the ball in ℝd, based on integrals of the function v(x), its normal derivatives or iterated Laplacians △ j v (j = 0,…, m i−1) over n distinct (d−1)dimensional hyperspheres S(ρ i )of radius ρ i , respectively, which integrate exactly wide classes of functions defined as the harmonic span of given radial functions, and ...
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Cubature Formula

2019
Masanori Sawa, Masatake Hirao, Sanpei Kageyama   +2 more
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