Mathematics and Its Applications

By: S. L. Sobolev, V. L. Vaskevich, Vladimir L. Vaskevich

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Hardcover | 30 June 1997

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444 Pages

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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.

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Non-Fiction Mathematics Calculus & Mathematical Analysis Numerical Analysis Probability & Statistics Functional Analysis & Transforms Real Analysis

Foreword to the English Translation
Preface
From the Preface to "Introduction to the Theory of Cubature Formulas"
Problems and Results of the Theory of Cubature Formulas	p. 1
Cubature Formulas of Finite Order	p. 43
Formulas with Regular Boundary Layer for Rational Polyhedra	p. 74
The Rate of Convergence of Cubature Formulas	p. 93
Cubature Formulas with Regular Boundary Layer	p. 131
Universal Asymptotic Optimality	p. 173
Cubature Formulas of Infinite Order	p. 221
Functions of a Discrete Variable	p. 292
Optimal Formulas	p. 331
References	p. 389
Notation Index	p. 411
Subject Index	p. 413
Table of Contents provided by Blackwell. All Rights Reserved.

Mathematics and Its Applications

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