Real Analysis with an Introduction to Wavelets and Applications (eBook)

Front Cover 1
Real Analysis with an Introduction to Wavelets and Applications 4
Copyright Page 5
Contents 8
Preface 14
Chapter I. Fundamentals 18
1 Elementary Set Theory 18
2 Relations and Orderings 23
3 Cardinality and Countability 30
4 The Topology of Rn 36
Chapter 2. Measure Theory 50
1 Classes of Sets 52
2 Measures on a Ring 56
3 Outer Measures and Lebesgue Measure 62
4 Measurable Functions 71
5 Convergence of Measurable Functions 78
Chapter 3. The Lebesgue integral 82
1 Riemann Integral and Lebesgue Integral 82
2 The General Lebesgue Integral 89
3 Convergence and Approximation of Lebesgue Integrals 94
4 Lebesgue Integrals in the Plane 100
Chapter 4. Special Topics of Lebesgue Integral and Applications 106
1 Differentiation and Integration 106
2 Mathematical Models for Probability 119
3 Convergence and Limit Theorems 126
Chapter 5. Vector Spaces, Hilbert Spaces, and the L2 Space 132
1 Groups, Fields, and Vector Spaces 132
2 Inner Product Spaces 142
3 The Space L2 148
4 Projections and Hilbert Space Isomorphisms 153
5 Banach Spaces 162
Chapter 6. Fourier Analysis 172
1 Fourier Series 173
2 Parseval's Formula 185
3 The Fourier Transform of Integrable Functions 193
4 Fourier Transforms of Square Integrable Functions 207
5 The Poisson Summation Formula 215
Chapter 7. Orthonormal Wavelet Bases 226
1 Haar Wavelet Basis 228
2 Multiresolution Analysis 236
3 Orthonormal Wavelets from MRA 243
4 Orthonormal Spline Wavelets 254
5 Fast Wavelet Transforms 265
6 Biorthogonal Wavelet Bases 277
Chapter 8. Compactly Supported Wavelets 288
1 Symbols of Orthonormal Scaling Functions 288
2 The Daubechies Scaling Functions 300
3 Computation of Daubechies Scaling Fimctions 309
4 Wavelet Packets 318
5 Compactly Supported Biorthogonal Wavelet Bases 325
Chapter 9. Wavelets In Signal Processing 332
1 Signals 332
2 Filters 341
3 Coding Signals by Wavelet Transform 354
4 Filter Banks 363
Appendix 370
Biliography 374
Index 378