The Theory of Cubature Formulas

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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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.

• Numerical integration
• cubature
• functional analysis
• numerical analysis

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

  S. L. Sobolev, V. L. Vaskevich

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