Theoretical Numerical Analysis: A Functional Analysis Framework

✍ Scribed by Kendall Atkinson, Weimin Han

• Publisher: Springer
• Year: 2007
• Tongue: English
• Leaves: 595
• Series: Texts in Applied Mathematics
• Edition: 2nd
• Category: Library

✦ Synopsis

This textbook covers basic results of functional analysis and also some additional topics which are needed in theoretical numerical analysis. For this second edition, a new chapter on Fourier analysis and wavelets and over 140 new exercises have been added, almost doubling the exercise amount from the last edition. Many sections from the first edition have been revised. Some of the other topics covered in this book are functional analysis and approximation theory, nonlinear analysis, Sobolev spaces, elliptic boundary value problems and variational inequalities.

✦ Table of Contents

Series Preface......Page 7
Preface to the Second Edition......Page 9
Preface to the First Edition......Page 11
Contents......Page 13
1.1 Linear spaces......Page 19
1.2 Normed spaces......Page 25
1.3 Inner product spaces......Page 39
1.4 Spaces of continuously di.erentiable functions......Page 56
1.5 L^p spaces......Page 61
1.6 Compact sets......Page 65
2 Linear Operators on Normed Spaces......Page 69
2.1 Operators......Page 70
2.2 Continuous linear operators......Page 73
2.3 The geometric series theorem and its variants......Page 78
2.4 Some more results on linear operators......Page 89
2.5 Linear functionals......Page 97
2.6 Adjoint operators......Page 103
2.7 Types of convergence......Page 108
2.8 Compact linear operators......Page 111
2.9 The resolvent operator......Page 125
3 Approximation Theory......Page 131
3.1 Approximation of continuous functions by polynomials......Page 132
3.2 Interpolation theory......Page 133
3.3 Best approximation......Page 147
3.4 Best approximations in inner product spaces, projection on closed convex sets......Page 157
3.5 Orthogonal polynomials......Page 164
3.6 Projection operators......Page 168
3.7 Uniform error bounds......Page 172
4.1 Fourier series......Page 179
4.2 Fourier transform......Page 193
4.3 Discrete Fourier transform......Page 198
4.4 Haar wavelets......Page 203
4.5 Multiresolution analysis......Page 211
5 Nonlinear Equations and Their Solution by Iteration......Page 219
5.1 The Banach .xed-point theorem......Page 220
5.2 Applications to iterative methods......Page 224
5.3 Di.erential calculus for nonlinear operators......Page 237
5.4 Newton’s method......Page 248
5.5 Completely continuous vector .elds......Page 254
5.6 Conjugate gradient method for operator equations......Page 257
6.1 Finite di.erence approximations......Page 267
6.2 Lax equivalence theorem......Page 274
6.3 More on convergence......Page 283
7.1 Weak derivatives......Page 291
7.2 Sobolev spaces......Page 297
7.3 Properties......Page 307
7.4 Characterization of Sobolev spaces via the Fourier transform......Page 321
7.5 Periodic Sobolev spaces......Page 325
7.6 Integration by parts formulas......Page 337
8 Variational Formulations of Elliptic Boundary Value Problems......Page 341
8.1 A model boundary value problem......Page 342
8.2 Some general results on existence and uniqueness......Page 344
8.3 The Lax-Milgram Lemma......Page 348
8.4 Weak formulations of linear elliptic boundary value problems......Page 352
8.5 A boundary value problem of linearized elasticity......Page 361
8.6 Mixed and dual formulations......Page 366
8.7 Generalized Lax-Milgram Lemma......Page 371
8.8 A nonlinear problem......Page 373
9.1 The Galerkin method......Page 379
9.2 The Petrov-Galerkin method......Page 385
9.3 Generalized Galerkin method......Page 388
9.4 Conjugate gradient method: variational formulation......Page 390
10 Finite Element Analysis......Page 395
10.1 One-dimensional examples......Page 397
10.2 Basics of the .nite element method......Page 405
10.3 Error estimates of .nite element interpolations......Page 414
10.4 Convergence and error estimates......Page 422
11.1 Introductory examples......Page 431
11.2 Elliptic variational inequalities of the .rst kind......Page 438
11.3 Approximation of EVIs of the .rst kind......Page 443
11.4 Elliptic variational inequalities of the second kind......Page 446
11.5 Approximation of EVIs of the second kind......Page 452
12 Numerical Solution of Fredholm Integral Equations of the Second Kind......Page 465
12.1 Projection methods: General theory......Page 466
12.2 Examples......Page 474
12.3 Iterated projection methods......Page 486
12.4 The Nystr¨om method......Page 496
12.5 Product integration......Page 510
12.6 Iteration methods......Page 522
12.7 Projection methods for nonlinear equations......Page 533
13 Boundary Integral Equations......Page 541
13.1 Boundary integral equations......Page 542
13.2 Boundary integral equations of the second kind......Page 555
13.3 A boundary integral equation of the .rst kind......Page 567
References......Page 573
Index......Page 587