Real Analysis with an Introduction to Wavelets and Applications,9780123548610

Real Analysis with an Introduction to Wavelets and Applications

by ; ;
ISBN13: 9780123548610
ISBN10: 0123548616
Format: Hardcover
Pub. Date: 2004-12-14
Publisher(s): Elsevier Science
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Summary

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

Table of Contents

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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