The Theory of Cubature Formulas / Edition 1

by S.L. Sobolev, Vladimir L. Vaskevich

Overview

This volume considers various methods for constructing cubature and quadrature formulas of arbitrary degree. These formulas are intended to approximate the calculation of multiple and conventional integrals over a bounded domain of integration. The latter is assumed to have a piecewise-smooth boundary and to be arbitrary in other aspects. Particular emphasis is placed on invariant cubature formulas and those for a cube, a simplex, and other polyhedra. Here, the techniques of functional analysis and partial differential equations are applied to the classical problem of numerical integration, to establish many important and deep analytical properties of cubature formulas. The prerequisites of the theory of many-dimensional discrete function spaces and the theory of finite differences are concisely presented. Special attention is paid to constructing and studying the optimal cubature formulas in Sobolev spaces. As an asymptotically optimal sequence of cubature formulas, a many-dimensional abstraction of the Gregory quadrature is indicated.
Audience: This book is intended for researchers having a basic knowledge of functional analysis who are interested in the applications of modern theoretical methods to numerical mathematics.

Product Details

ISBN-13:	9780792346319
Publisher:	Springer Netherlands
Publication date:	06/30/1997
Series:	Mathematics and Its Applications , #415
Edition description:	1997
Pages:	418
Product dimensions:	6.10(w) x 9.25(h) x 0.04(d)

Table of Contents

Foreword to the English Translation. Preface. From the Preface to ‘Introduction to the Theory of Cubature Formulas'. 1. Problems and Results of the Theory of Cubature Formulas. 2. Cubature Formulas of Finite Order. 3. Formulas with Regular Boundary Layer for Rational Polyhedra. 4. The Rate of Convergence of Cubature Formulas. 5. Cubature Formulas with Regular Boundary Layer. 6. Universal Asymptotic Optimality. 7. Cubature Formulas of Infinite Order. 8. Functions of A Discrete Variable. 9. Optimal Formulas. References. Notation Index. Subject Index.

Editorial Reviews

Considers various methods for constructing cubature and quadrature formulas of arbitrary degree, with particular emphasis on invariant cubature formulas and those for a cube, a simplex, and other polyhedra. After establishing important analytical properties of cubature formulas, the prerequisites of the theory of many-dimensional discrete function spaces and the theory of finite differences are concisely presented. Special attention is paid to constructing and studying the optimal cubature formulas in Sobolev spaces. Annotation c. by Book News, Inc., Portland, Or.

Booknews