Real Analysis with an Introduction to

An in-depth look at real analysis and its applications, including an introduction to wavelet analysis, a popular topic in "applied real analysis". This text makes a very natural connection between the classic pure analysis and the applied topics, including measure theory, Lebesgue Integral, harmonic analysis and wavelet theory with many associated applications. *The text is relatively elementary at the start, but the level of difficulty steadily increases *The book contains many clear, detailed examples, case studies and exercises *Many real world applications relating to measure theory and pure analysis *Introduction to wavelet analysis

Preface

xiii

Chapter 1 Fundamentals

1 Elementary Set Theory

1

(5)

2 Relations and Orderings

6

(7)

3 Cardinality and Countability

13

(6)

4 The Topology of Rn

19

(14)

Chapter 2 Measure Theory

33

(32)

1 Classes of Sets

35

(4)

2 Measures on a Ring

39

(6)

3 Outer Measures and Lebesgue Measure

45

(9)

4 Measurable Functions

54

(7)

5 Convergence of Measurable Functions

61

(4)

Chapter 3 The Lebesgue Integral

65

(24)

1 Riemann Integral and Lebesgue Integral

65

(7)

2 The General Lebesgue Integral

72

(5)

3 Convergence and Approximation of Lebesgue Integrals

77

(6)

4 Lebesgue Integrals in the Plane

83

(6)

Chapter 4 Special Topics of Lebesgue Integral and Applications

89

(26)

1 Differentiation and Integration

89

(13)

2 Mathematical Models for Probability

102

(7)

3 Convergence and Limit Theorems

109

(6)

Chapter 5 Vector Spaces, Hilbert Spaces, and the L² Space

115

(40)

1 Groups, Fields, and Vector Spaces

115

(10)

2 Inner Product Spaces

125

(6)

3 The Space L²

131

(5)

4 Projections and Hilbert Space Isomorphisms

136

(9)

5 Banach Spaces

145

(10)

Chapter 6 Fourier Analysis

155

(54)

1 Fourier Series

156

(12)

1.1 The Finite Fourier Transform and Its Properties

157

(4)

1.2 Convergence of Fourier Series

161

(2)

1.3 The Study of Functions Using Fourier Series

163

(5)

2 Parseval's Formula

168

(8)

2.1 The Space of Square Integrable Periodic Functions

168

(2)

2.2 The Convolution Theorem

170

(1)

2.3 Parseval's Formula

171

(2)

2.4 The Finite Fourier Transform on General Intervals

173

(3)

3 The Fourier Transform of Integrable Functions

176

(14)

3.1 Definition and Properties

176

(3)

3.2 The Convolution Theorem

179

(2)

3.3 The Inverse Fourier Transform

181

(2)

3.4 The Study of Functions Using Fourier Transforms

183

(4)

3.5 Proof of Theorem 6.3.4

187

(3)

4 Fourier Transforms of Square Integrable Functions

190

(8)

4.1 Definition and Properties

191

(3)

4.2 Plancherel's Theorem

194

(2)

4.3 The Fourier Transform of Derivatives

196

(2)

5 The Poisson Summation Formula

198

(11)

5.1 The Poisson Summation Formula for L¹

198

(5)

5.2 Fourier Transforms of Compactly Supported Functions

203

(3)

5.3 The Poisson Summation Formula for Compactly Supported Functions

206

(3)

Chapter 7 Orthonormal Wavelet Bases

209

(62)

1 Haar Wavelet Basis

211

(8)

1.1 Approximation by Step Functions

211

(1)

1.2 The Haar Wavelet Basis

212

(3)

1.3 The Decomposition of Functions into Haar Wavelet Series

215

(4)

2 Multiresolution Analysis

219

(7)

2.1 Definition of Multiresolution Analysis

219

(2)

2.2 Stability of Scaling Functions

221

(2)

2.3 Completeness of Scaling Functions

223

(3)

3 Orthonormal Wavelets from MRA

226

(11)

3.1 Orthonormalization

227

(4)

3.2 Orthonormal Wavelets

231

(6)

4 Orthonormal Spline Wavelets

237

(11)

4.1 Cardinal B-splines

237

(4)

4.2 Construction of Orthonormal Spline Wavelets

241

(7)

5 Fast Wavelet Transforms

248

(12)

5.1 Initialization

248

(3)

5.2 Multiscale Decomposition

251

(1)

5.3 The Fast Wavelet Transform

252

(2)

5.4 Pyramid Algorithms

254

(2)

5.5 Approximation of Functions by Wavelet Series

256

(4)

6 Biorthogonal Wavelet Bases

260

(11)

6.1 Construction of Biorthogonal Wavelet Bases

261

(5)

6.2 Decomposition and Recovering of Functions

266

(5)

Chapter 8 Compactly Supported Wavelets

271

(44)

1 Symbols of Orthonormal Scaling Functions

271

(12)

1.1 Basic Properties of the Mask

272

(2)

1.2 The Symbol of an Orthonormal Scaling Function

274

(9)

2 The Daubechies Scaling Functions

283

(9)

2.1 The Infinite Product Form

284

(3)

2.2 Proof That φLepsilonL²

287

(1)

2.3 Orthogonality of φL

288

(4)

3 Computation of Daubechies Scaling Functions

292

(9)

3.1 The Cascade Algorithm

292

(4)

3.2 The Recursion Algorithm

296

(3)

3.3 Convergence of the Cascade Algorithm

299

(2)

4 Wavelet Packets

301

(7)

4.1 The Construction of Wavelet Packets

301

(3)

4.2 Orthonormal Bases from Wavelet Packets

304

(4)

5 Compactly Supported Biorthogonal Wavelet Bases

308

(7)

5.1 Symmetry of the Scaling Function and Its Mask

308

(3)

5.2 The Construction of Symmetric Biorthogonal Scaling Functions and Wavelets

311

(4)

Chapter 9 Wavelets in Signal Processing

315

(38)

1 Signals

315

(9)

1.1 Analog Signals

316

(1)

1.2 Approximation

317

(2)

1.3 Sampling Theorems

319

(2)

1.4 Discrete Signals

321

(3)

2 Filters

324

(13)

2.1 Representing Filters in the Time Domain

326

(2)

2.2 Filters in the Frequency Domain

328

(1)

2.3 Lowpass Filters and Highpass Filters

329

(2)

2.4 Dual Filters

331

(1)

2.5 Magnitude and Phase

332

(2)

2.6 Inverse Filters

334

(3)

3 Coding Signals by Wavelet Transform

337

(9)

3.1 Coding Signals Using Shannon Wavelets

337

(3)

3.2 Alias Cancellation

340

(1)

3.3 Coding Signals Using Other Wavelets

341

(2)

3.4 Sampling Data Coding

343

(3)

4 Filter Banks

346

(7)

4.1 Conditions for Biorthogonal Filter Banks

348

(5)

Appendix

353

(4)

Bibliography

357

(4)

Index

361

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Real Analysis with an Introduction to Wavelets and Applications

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Table of Contents

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