On best cubature formulas and spline interpolation

Summary A general cubature formula with an arbitrary preassigned weight function is derived using monosplines and integration by parts. The problem of determining the best cubature is formulated in terms of monosplines of least deviation and a solution to the problem is given by Theorem 3 below. This theorem may also be viewed as an optimal property of a new kind of two-dimensional spline interpolation.This work was done while the author was working at CERN, Geneva, Switzerland     

On the accuracy of cubature formulas in Sobolev spaces

We establish the necessary and sufficient conditions for the boundedness of the cubature formulas error functionals in spaces of type L _p ^m corresponding to the considered sets of integrable functions defined on bounded subsets of cylindrical and conical surfaces.     

On criteria of estimation of the errors of cubature formulas

The paper considers cubature formulas for calculating integrals of functions f(X), X = (x ₁, …, x _n ) which are defined on the n-dimensional unit hypercube K ⁿ = [0, 1] ⁿ and have integrable mixed derivatives of the kind \(\partial _{\begin{array}{*{20}c} {\alpha _1 \alpha _n } \\ {x_1 , \ldots , x_n } \\ \end{array} } f(X)\), 0 ≤ α _j ≤ 2. We estimate the errors R[f] = \(\smallint _{K^n } \) f(X)dX ? Σ _{k = 1} ^N c _k f(X(k)) of cubature formulas (c _k > 0) as functions of the weights c _k of nodes X(k) and properties of integrable functions. The error is estimated in terms of the integrals of the derivatives of f over r-dimensional faces (r≤n) of the hypercube K ⁿ : |R(f)| ≤ \(\sum _{\alpha _j } \) G(α _j )\(\int_{K^r } {\left| {\partial _{\begin{array}{*{20}c} {\alpha _1 \alpha _n } \\ {x_1 , \ldots , x_n } \\ \end{array} } f(X)} \right|} \) dX _r , where coefficients G(α _j ) are criteria which depend only on parameters c _k and X(k). We present an algorithm to calculate these criteria in the two- and n-dimensional cases. Examples are given. A particular case of the criteria is the discrepancy, and the algorithm proposed is a generalization of those used to compute the discrepancy. The results obtained can be used for optimization of cubature formulas as functions of c _k and X(k).     

An encyclopaedia of cubature formulas

About 13 years ago we started collecting published cubature formulas for the approximation of multivariate integrals over some standard regions. In this paper we describe how we make this information available to a larger audience via the World Wide Web.     

On the construction of cubature formulas invariant under dihedral groups

We study cubature formulas invariant under the dihedral group of order 16p. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 7, pp. 983–991, July, 2008.     

Invariant cubature formulas for the hyperoctahedron

Cubature formulas for calculating integrals over the hyperoctahedron that are invariant under the group of all of its orthogonal transformations are obtained. Two of them are exact for all polynomials of degree no greater than seven and one is exact for all polynomials of degree no greater than five. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 734–741, May, 1997. Translated by V. N. Dubrovsky     

Optimal cubature formulas on compact homogeneous manifolds

We find lower bounds for the rate of convergence of optimal cubature formulas on sets of differentiable functions on compact homogeneous manifolds of rank I or two-point homogeneous spaces. It is shown that these lower bounds are sharp in the power scale in the case of S², the unit sphere in R³.     

The existence of Gaussian cubature formulas

We provide a necessary and sufficient condition for the existence of Gaussian cubature formulas. It consists of checking whether an overdetermined linear system has a solution and so complements Mysovskikh’s theorem which requires computing common zeros of orthonormal polynomials. Moreover, the size of the linear system shows that the existence of a cubature formula imposes severe restrictions on the associated linear functional. For fixed precision (or degree), the larger the number of variables, the worse it gets. And for fixed number of variables, the larger the precision, the worse it gets. Finally, we also provide an interpretation of the necessary and sufficient condition in terms of the existence of a polynomial with very specific properties.     

Asymptotically optimal cubature weight formulas

Krasnoyarsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 30, No. 2, pp. 151–160, March–April, 1989.     

Asymptotically best sequences of cubature formulas

Siberian Mathematical Journal -     

Spherical cubature formulas in Sobolev spaces

We study sequences of cubature formulas on the unit sphere in a multidimensional Euclidean space. The grids for the cubature formulas under consideration embed in each other consecutively, forming in the limit a dense subset on the initial sphere. As the domain of cubature formulas, i.e. as the class of integrands, we take spherical Sobolev spaces. These spaces may have fractional smoothness. We prove that, among all possible spherical cubature formulas with given grid, there exists and is unique a formula with the least norm of the error, an optimal formula. The weights of the optimal cubature formula are shown to be solutions to a special nondegenerate system of linear equations. We prove that the errors of cubature formulas tend to zero as the number of nodes grows indefinitely.     

Weighted cubature formulas for periodic functions

This article contains a study of weighted cubature formulas for periodic functions in n-dimensional Euclidean space with matrix period H. The principal error terms are considered.Translated from Matematicheskie Zametki, Vol. 3, No. 3, pp. 319–326, March, 1968.The author wishes to express his deep appreciation of S. L. Sobolev's guidance in this work.