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The Theory of Cubature Formulas

  • Book
  • © 1997

Overview

Authors:
  1. S. L. Sobolev

    1. Sobolev Institute of Mathematics, Siberian Division of the Russian Academy of Sciences, Novosibirsk, Russia

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  2. V. L. Vaskevich

    1. Sobolev Institute of Mathematics, Siberian Division of the Russian Academy of Sciences, Novosibirsk, Russia

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    You can also search for this author in PubMed   Google Scholar

Part of the book series: Mathematics and Its Applications (MAIA, volume 415)

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Table of contents (9 chapters)

  1. Front Matter

    Pages i-ix
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  2. Problems and Results of the Theory of Cubature Formulas

    • S. L. Sobolev, V. L. Vaskevich
    Pages 1-42

  3. Cubature Formulas of Finite Order

    • S. L. Sobolev, V. L. Vaskevich
    Pages 43-73

  4. Formulas with Regular Boundary Layer for Rational Polyhedra

    • S. L. Sobolev, V. L. Vaskevich
    Pages 74-92

  5. The Rate of Convergence of Cubature Formulas

    • S. L. Sobolev, V. L. Vaskevich
    Pages 93-130

  6. Cubature Formulas with Regular Boundary Layer

    • S. L. Sobolev, V. L. Vaskevich
    Pages 131-172

  7. Universal Asymptotic Optimality

    • S. L. Sobolev, V. L. Vaskevich
    Pages 173-220

  8. Cubature Formulas of Infinite Order

    • S. L. Sobolev, V. L. Vaskevich
    Pages 221-291

  9. Functions of a Discrete Variable

    • S. L. Sobolev, V. L. Vaskevich
    Pages 292-330

  10. Optimal Formulas

    • S. L. Sobolev, V. L. Vaskevich
    Pages 331-388

  11. Back Matter

    Pages 389-418
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Keywords

About this book

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.

Authors and Affiliations

  • Sobolev Institute of Mathematics, Siberian Division of the Russian Academy of Sciences, Novosibirsk, Russia

    S. L. Sobolev, V. L. Vaskevich

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