理论数值分析 : 第3版

Series PrefacePrefaceLinear Spaces1.1 Linear spaces1.2 Normed spaces1.2.1 Convergence1.2.2 Banach spaces1.2.3 Completion of normed spaces1.3 Inner product spaces1.3.1 Hilbert spaces1.3.2 Orthogonality1.4 Spaces of continuously differentiable functions1.4.1 HSlder spaces1.5 Lp spaces1.6 Compact setsLinear Operators on Normed Spaces2.1 Operators2.2 Continuous linear operators2.2.1 (V,W) as a Banach space2.3 The geometric series theorem and its variants2.3.1 A generalization2.3.2 A perturbation result2.4 Some more results on linear operators2.4.1 An extension theorem2.4.2 Open mapping theorem2.4.3 Principle of uniform boundedness2.4.4 Convergence of numerical quadratures2.5 Linear functionals2.5.1 An extension theorem for linear functionals2.5.2 The Riesz representation theorem2.6 Adjoint operators2.7 Weak convergence and weak compactness2.8 Compact linear operators2.8.1 Compact integral operators on C(D)2.8.2 Properties of compact operators2.8.3 Integral operators on L2(a,b)2.8.4 The Fredholm alternative theorem2.8.5 Additional results on Fredholm integral equations2.9 The resolvent operator2.9.1 R(A) as a holomorphic functionApproximation Theory3.1 Approximation of continuous functions by polynomials3.2 Interpolation theory3.2.1 Lagrange polynomial interpolation3.2.2 Hermite polynomial interpolation3.2.3 Piecewise polynomial interpolation3.2.4 Trigonometric interpolation3.3 Best approximation3.3.1 Convexity,lower semicontinuity3.3.2 Some abstract existence results3.3.3 Existence of best approximation3.3.4 Uniqueness of best approximation3.4 Best approximations in inner product spaces,projection onclosed convex sets3.5 Orthogonal polynomials3.6 Projection operators3.7 Uniform error bounds3.7.1 Uniform error bounds for L2-approximations3.7.2 L2-approximations using polynomials3.7.3 Interpolatory projections and their convergenceFourier Analysis and Wavelets4.1 Fourier series4.2 Fourier transform4.3 Discrete Fourier transform4.4 Haar wavelets4.5 Multiresolution analysisNonlinear Equations and Their Solution by Iteration5.1 The Banach fixed-point theorem5.2 Applications to iterative methods5.2.1 Nonlinear algebraic equations5.2.2 Linear algebraic systems5.2.3 Linear and nonlinear integral equations5.2.4 Ordinary differential equations in Banach spaces5.3 Differential calculus for nonlinear operators5.3.1 Frechet and Gateaux derivatives5.3.2 Mean value theorems5.3.3 Partial derivatives5.3.4 The Gateaux derivative and convex minimization5.4 Newton's method5.4.1 Newton's method in Banach spaces5.4.2 Applications5.5 Completely continuous vector fields5.5.1 The rotation of a completely continuous vector field5.6 Conjugate gradient method for operator equationsFinite Difference Method6.1 Finite difference approximations6.2 Lax equivalence theorem6.3 More on convergenceSobolev Spaces7.1 Weak derivatives7.2 Sobolev spaces7.2.1 Sobolev spaces of integer order7.2.2 Sobolev spaces of real order7.2.3 Sobolev spaces over boundaries7.3 Properties7.3.1 Approximation by smooth functions7.3.2 Extensions7.3.3 Sobolev embedding theorems7.3.4 Traces7.3.5 Equivalent norms7.3.6 A Sobolev quotient space7.4 Characterization of Sobolev spaces via the Fouriertransform7.5 Periodic Sobolev spaces7.5.1 The dual space7.5.2 Embedding results7.5.3 Approximation results7.5.4 An illustrative example of an operator7.5.5 Spherical polynomials and spherical harmonics7.6 Integration by parts formulas8 Weak Formulations of Elliptic Boundary Value Problems8.1 A model boundary value problem8.2 Some general results on existence and uniqueness8.3 The Lax-Milgram Lemma8.4 Weak formulations of linear elliptic boundary valueproblems8.4.1 Problems with homogeneous Dirichlet boundarycon-ditions8.4.2 Problems with non-homogeneous Dirichlet boundaryconditions8.4.3 Problems with Neumann boundary conditions8.4.4 Problems with mixed boundary conditions8.4.5 A general linear second-order elliptic boundary valueproblem8.5 A boundary value problem of linearized elasticity8.6 Mixed and dual formulations8.7 Generalized Lax-Milgram Lemma8.8 A nonlinear problem9 The Galerkin Method and Its Variants9.1 The Galerkin method9.2 The Petrov-Galerkin method9.3 Generalized Galerkin method9.4 Conjugate gradient method： variational formulation10 Finite Element Analysis10.1 One-dimensional examples10.1.1 Linear elements for a second-order problem10.1.2 High order elements and the condensation technique10.1.3 Reference element technique10.2 Basics of the finite element method10.2.1 Continuous linear elements10.2.2 Affine-equivalent finite elements10.2.3 Finite element spaces10.3 Error estimates of finite element interpolations10.3.1 Local interpolations10.3.2 Interpolation error estimates on the reference element10.3.3 Local interpolation error estimates10.3.4 Global interpolation error estimates10.4 Convergence and error estimates……11 Elliptic Variational Inequalities and Their NumericalAp-proximations12 Numerical Solution of Fredholm Integral Equations of the SecondKind13 Boundary Integral Equations14 Multivariable Polynomial ApproximationsReferencesIndex

　　《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。