Sobolev S.L., Vaskevich V.L. The Theory of Cubature Formulas

Springer, 2011. — 428 p.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.Foreword to the English Translation
Problems and Results of the Theory of Cubature Formulas
Exact Formulas
Functional-Analytical Statement of the Problem
The Order of a Cubature Formula on Infinitely Differentiable Functions
Errors in W2(m)
Expansion of the L2(m) Norm of an Error with Arbitrary Nodes
The Weights of Optimal Cubature Formulas on a Given Lattice
Cubature Formulas of Finite Order
Formulas of Interpolatory Type
Rotation Invariant Cubature Formulas
Rotation Invariant Cubature Formulas on the Sphere in R^3
Formulas with Regular Boundary Layer for Rational Polyhedra
Rational Polyhedra
Constructing Formulas for Rational Polyhedra
A Formal Boundary Layer
The Rate of Convergence of Cubature Formulas
A Universal Lower Bound on the Rate of Convergence
The Rate of Convergence of a Homogeneous Error
The Bakhvalov Theorem
The Rate of Convergence of an Equidistributed Error
Cubature Formulas with Regular Boundary Layer
The Properties of the Extremal Function of an L2(m)-Optimal Error
Errors in the L2(m)(Ω) Space of Compactly-Supported Functions
Constructing a Formula with Regular Boundary Layer
Asymptotic Expansion of the Norm of an Error with Regular Boundary Layer
The Properties of the Extremal Function of an Error in L2(m)(Ω)
Universal Asymptotic Optimality
Cubature Formulas with Bounded Boundary Layer in Hilbert Spaces
Cubature Formulas with Bounded Boundary Layer in Holder Spaces
Constructing Universal Asymptotically Optimal Formulas
Cubature Formulas of Infinite Order
Weak Convergence of Cubature Formulas
The Function Classes H(ϰ, A, λ) and C(ϰ, A, λ)
The Properties of H(ϰ, A) and C(ϰ, A) for ϰ > 1
The Function Classes Ψ(ρ, σ, μ)
The Classes of Periodic Functions H(ϰ, A, λ)
Convergence of Cubature Formulas in H(ϰ, A, λ)
Gevrey Classes of Functions in a Single Independent Variable
Convergence of Euler-Maclaurin and Gregory Quadrature Formulas on Gevrey Classes
The Sequence of the Fourier Coefficients of an Error
The Fourier Transform of a Local Error
The Norm of the Error of a Gregory Quadrature Formula
Functions of a Discrete Variable
Operations over Discrete Functions
Spaces of Discrete Functions
The Fourier Transform of a Discrete Function
Optimal Formulas
Statement of the Problem of Optimal Weights
The Fundamental Solution to the Convolution Equation
A Discrete Analog of the Polyharmonic Operator
The Weights of Optimal Formulas and the Extension Problem
A One-Dimensional Discrete Analog of a Derivative of Even Order
The Roots of the Euler Polynomial
The First Asymptotic Formula
The Weights of Optimal Quadrature Formulas
References
Notation Index
Subject Index