On expansions of functions of two variables in mixed Fourier-Jacobi series and their application for estimation of errors of cubature formulas

Some problems of expansion of functions of two variables in mixed Fourier-Jacobi series are discussed. In particular, estimates of the convergence rate of these series on classes of functions of two variables characterized by generalized moduli of continuity are given. Applications of these results and estimation of residues of some Chebyshev-type mixed cubature formulas are discussed.     

On cubature formulas on the basis of lattices

Q (.. , L). Q . P(S_r(2)) — 2 (S _r(2) (r — ). , M(P(S _r(m=sup{t(·)t(·)₁:t P(S _r(2)),t 0}. , /4+(1)M(P(S _r(2)))/r ²15/17+(1)(r+). (Q), Q L.     

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 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.     

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.     

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     

Asymptotically optimal cubature weight formulas

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

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 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.     

Asymptotically exact estimation of the error of weighted cubature formulas optimal in some classes of continuous functions

We obtain asymptotically exact estimates of the error of weighted cubature formulas optimal in the classes of functions satisfying the Hölder condition in the case where the weight function is nonnegative and Lebesgue integrable.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 10, pp. 1405–1415, October, 1995.