《理论数值分析(第3版)》旨在为读者提供一个基于泛函分析并专注于数值分析的数学框架，让读者更好地学习数值分析和计算数学，及早进入科研项目。本教程包括了泛函分析、逼近理论、傅里叶分析和小波等诸多基础专题，每个专题的表述既能了解该科目，又可以达到一定的深度，特别专题的参考文献都列于每章末，供读者深入学习和研究。由于现实问题的往往是多相关的，多变量多项式在研究和应用中扮演着重要的角色，第三版中就此专题新增了一章。本书较前一版本有两章做了重大修改；整本书自始至终都有很多小改动，并且增加了不少新练习。

本书由阿特肯森著。

Series Preface

Preface

1 Linear Spaces

 1.1 Linear spaces

 1.2 Normed spaces

 1.2.1 Convergence

 1.2.2 Banach spaces

 1.2.3 Completion of normed spaces

 1.3 Inner product spaces

 1.3.1 Hilbert spaces

 1.3.2 Orthogonality

 1.4 Spaces of continuously differentiable functions

 1.4.1 HSlder spaces

 1.5 Lp spaces

 1.6 Compact sets

2 Linear Operators on Normed Spaces

 2.1 Operators

 2.2 Continuous linear operators

 2.2.1 C(V, W) as a Banach space

 2.3 The geometric series theorem and its variants

 2.3.1 A generalization

 2.3.2 A perturbation result

 2.4 Some more results on linear operators

 2.4.1 An extension theorem

 2.4.2 Open mapping theorem

 2.4.3 Principle of uniform boundedness

 2.4.4 Convergence of numerical quadratures

 2.5 Linear functionals

 2.5.1 An extension theorem for linear functionals

 2.5.2 The Riesz representation theorem

 2.6 Adjoint operators

 2.7 Weak convergence and weak compactness

 2.8 Compact linear operators

 2.8.1 Compact integral operators on C(D)

 2.8.2 Properties of compact operators

 2.8.3 Integral operators on L2(a,b)

 2.8.4 The Fredholm alternative theorem

 2.8.5 Additional results on Fredholm integral equations

 2.9 The resolvent operator

 2.9.1 R(r) as a holomorphic function

3 Approximation Theory

 3.1 Approximation of continuous functions by polynomials

 3.2 Interpolation theory

 3.2.1 Lagrange polynomial interpolation

 3.2.2 Hermite polynomial interpolation

 3.2.3 Piecewise polynomial interpolation

 3.2.4 Trigonometric interpolation

 3.3 Best approximation

 3.3.1 Convexity, lower semicontinuity

 3.3.2 Some abstract existence results

 3.3.3 Existence of best approximation

 3.3.4 Uniqueness of best approximation

 3.4 Best approximations in inner product spaces, projection or closed convex sets

 3.5 Orthogonal polynomials

 3.6 Projection operators

 3.7 Uniform error bounds

 3.7.1 Uniform error bounds for L2-approximations

 3.7.2 L2-approximations using polynomials

 3.7.3 Interpolatory projections and their convergence

4 Fourier Analysis and Wavelets

 4.1 Fourier series

 4.2 Fourier transform

 4.3 Discrete Fourier transform

 4.4 Haar wavelets

 4.5 Multiresolution analysis

5 Nonlinear Equations and Their Solution by Iteration

 5.1 The Banach fixed-point theorem

 5.2 Applications to iterative methods

 5.2.1 Nonlinear algebraic equations

 5.2.2 Linear algebraic systems

 5.2.3 Linear and nonlinear integral equations

 5.2.4 Ordinary differential equations in Banach spaces

 5.3 Differential calculus for nonlinear operators

 5.3.1 Frechet and Ggteaux derivatives

 5.3.2 Mean value theorems

 5.3.3 Partial derivatives

 5.3.4 The Gateaux derivative and convex minimization

 5.4 Newton's method

 5.4.1 Newton's method in Banach spaces

 5.4.2 Applications

 5.5 Completely continuous vector fields

 5.5.1 The rotation of a completely continuous vector field

 5.6 Conjugate gradient method for operator equations

6 Finite Difference Method

 6.1 Finite difference approximations

 6.2 Lax equivalence theorem

 6.3 More on convergence

7 Sobolev Spaces

 7.1 Weak derivatives

 7.2 Sobolev spaces

 7.2.1 Sobolev spaces of integer order

 7.2.2 Sobolev spaces of real order

 7.2.3 Sobolev spaces over boundaries

 7.3 Properties

 7.3.1 Approximation by smooth functions

 7.3.2 Extensions

 7.3.3 Sobolev embedding theorems

 7.3.4 Traces

 7.3.5 Equivalent norms

 7.3.6 A Sobolev quotient space

 7.4 Characterization of Sobolev spaces via the Fourier transform

 7.5 Periodic Sobolev spaces

 7.5.1 The dual space

 7.5.2 Embedding results

 7.5.3 Approximation results

 7.5.4 An illustrative example of an operator

 7.5.5 Spherical polynomials and spherical harmonics

 7.6 Integration by parts formulas

8 Weak Formulations of Elliptic Boundary Value Problems

 8.1 A model boundary value problem

 8.2 Some general results on existence and uniqueness

 8.3 The Lax-Milgram Lemma

 8.4 Weak formulations of linear elliptic boundary value problems

 8.4.1 Problems with homogeneous Dirichlet boundary con- ditions

 8.4.2 Problems with non-homogeneous Dirichlet boundary conditions

 8.4.3 Problems with Neumann boundary conditions

 8.4.4 Problems with mixed boundary conditions

 8.4.5 A general linear second-order elliptic boundary value problem

 8.5 A boundary value problem of linearized elasticity

 8.6 Mixed and dual formulations

 8.7 Generalized Lax-Milgram Lemma

 8.8 A nonlinear problem

9 The Galerkin Method and Its Variants

 9.1 The Galerkin method

 9.2 The Petrov-Galerkin method

 9.3 Generalized Galerkin method

 9.4 Conjugate gradient method: variational formulation

10 Finite Element Analysis

 10.1 One-dimensional examples

 10.1.1 Linear elements for a second-order problem

 10.1.2 High order elements and the condensation technique

 10.1.3 Reference element technique

 10.2 Basics of the finite element method

 10.2.1 Continuous linear elements

 10.2.2 Affine~equivalent finite elements

 10.2.3 Finite element spaces

 10.3 Error estimates of finite element interpolations

 10.3.1 Local interpolations

 10.3.2 Interpolation error estimates on the reference element

 10.3.3 Local interpolation error estimates

 10.3.4 Global interpolation error estimates

 10.4 Convergence and error estimates

11 Elliptic Variational Inequalities and Their Numerical Approximations

 11.1 From variational equations to variational inequalities

 11.2 Existence and uniqueness based on convex minimization

 11.3 Existence and uniqueness results for a family of EVIs

 11.4 Numerical approximations

 11.5 Some contact problems in elasticity

 11.5.1 A frictional contact problem

 11.5.2 A Signorini frictionless contact problem

12 Numerical Solution of Fredholm Integral Equations of the Second Kind

 12.1 Projection methods: General theory

 12.1.1 Collocation methods

 12.1.2 Galerkin methods

 12.1.3 A general theoretical framework

 12.2 Examples

 12.2.1 Piecewise linear collocation

 12.2.2 Trigonometric polynomial collocation

 12.2.3 A piecewise linear Galerkin method

 12.2.4 A Galerkin method with trigonometric polynomials

 12.3 Iterated projection methods

 12.3.1 The iterated Galerkin method

 12.3.2 The iterated collocation solution

 12.4 The Nystrom method

 12.4.1 The NystrSm method for continuous kernel functions

 12.4.2 Properties and error analysis of the NystrSm method

 12.4.3 Collectively compact operator approximations

 12.5 Product integration

 12.5.1 Error analysis

 12.5.2 Generalizations to other kernel functions

 12.5.3 Improved error results for special kernels

 12.5.4 Product integration with graded meshes

 12.5.5 The relationship of product integration and collocation methods

 12.6 Iteration methods

 12.6.1 A two-grid iteration method for the Nystrom method

 12.6.2 Convergence analysis

 12.6.3 The iteration method for the linear system

 12.6.4 An operations count

 12.7 Projection methods for nonlinear equations

 12.7.1 Linearization

 12.7.2 A homotopy argument

 12.7.3 The approximating finite-dimensional problem

13 Boundary Integral Equations

 13.1 Boundary integral equations

 13.1.1 Green's identities and representation formula

 13.1.2 The Kelvin transformation and exterior problems

 13.1.3 Boundary integral equations of direct type

 13.2 Boundary integral equations of the second kind

 13.2.1 Evaluation of the double layer potential

 13.2.2 The exterior Neumann problem

 13.3 A boundary integral equation of the first kind

 13.3.1 A numerical method

14 Multivariable Polynomial Approximations

 14.1 Notation and best approximation results

 14.2 0rthogonal polynomials

 14.2.1 Triple recursion relation

 14.2.2 The orthogonal projection operator and its error

 14.3 Hyperinterpolation

 14.3.1 The norm of the hyperinterpolation operator

 14.4 A Galerkin method for elliptic equations

 14.4.1 The Galerkin method and its convergence

References

Index