From Euclidean to Hilbert Spaces

From Euclidian to Hilbert Spaces analyzes the transition from finite dimensional Euclidian spaces to infinite-dimensional Hilbert spaces, a notion that can sometimes be difficult for non-specialists to grasp. The focus is on the parallels and differences between the properties of the finite and infinite dimensions, noting the fundamental importance of coherence between the algebraic and topological structure, which makes Hilbert spaces the infinite-dimensional objects most closely related to Euclidian spaces.

The common thread of this book is the Fourier transform, which is examined starting from the discrete Fourier transform (DFT), along with its applications in signal and image processing, passing through the Fourier series and finishing with the use of the Fourier transform to solve differential equations.

The geometric structure of Hilbert spaces and the most significant properties of bounded linear operators in these spaces are also covered extensively. The theorems are presented with detailed proofs as well as meticulously explained exercises and solutions, with the aim of illustrating the variety of applications of the theoretical results.

Auteur
Edoardo Provenzi is Professor of Mathematics at the University of Bordeaux, France. He studies visual phenomena and their applications in image processing and computer vision, employing tools from differential geometry, harmonic analysis and mathematical physics.

Contenu

Preface xi

Chapter 1. Inner Product Spaces (Pre-Hilbert) 1

1.1. Real and complex inner products 1

1.2. The norm associated with an inner product and normed vector spaces 6

1.2.1. The parallelogram law and the polarization formula 9

1.3. Orthogonal and orthonormal families in inner product spaces 11

1.4. Generalized Pythagorean theorem 11

1.5. Orthogonality and linear independence 13

1.6. Orthogonal projection in inner product spaces 15

1.7. Existence of an orthonormal basis: the Gram-Schmidt process 19

1.8. Fundamental properties of orthonormal and orthogonal bases 20

1.9. Summary 28

Chapter 2. The Discrete Fourier Transform and its Applications to Signal and Image Processing 31

2.1. The space l*2(*ZN) and its canonical basis 31

2.1.1. The orthogonal basis of complex exponentials in l*2(*ZN) 34

2.2. The orthonormal Fourier basis of l*2(*ZN) 38

2.3. The orthogonal Fourier basis of l*2(*ZN) 40

2.4. Fourier coefficients and the discrete Fourier transform 41

2.4.1. The inverse discrete Fourier transform 44

2.4.2. Definition of the DFT and the IDFT with the orthonormal Fourier basis 46

2.4.3. The real (orthonormal) Fourier basis 47

2.5. Matrix interpretation of the DFT and the IDFT 48

2.5.1. The fast Fourier transform 51

2.6. The Fourier transform in signal processing 51

2.6.1. Synthesis formula for 1D signals: decomposition on the harmonic basis 51

2.6.2. Signification of Fourier coefficients and spectrums of a 1D signal 53

2.6.3. The synthesis formula and Fourier coefficients of the unit pulse 54

2.6.4. High and low frequencies in the synthesis formula 55

2.6.5. Signal filtering in frequency representation 59

2.6.6. The multiplication operator and its diagonal matrix representation 60

2.6.7. The Fourier multiplier operator 60

2.7. Properties of the DFT 61

2.7.1. Periodicity of and ? 62

2.7.2. DFT and shift 63

2.7.3. DFT and conjugation 67

2.7.4. DFT and convolution 68

2.8. The DFT and stationary operators 73

2.8.1. The DFT and the diagonalization of stationary operators 75

2.8.2. Circulant matrices 77

2.8.3. Exhaustive characterization of stationary operators 78

2.8.4. High-pass, low-pass and band-pass filters 82

2.8.5. Characterizing stationary operators using shift operators 83

2.8.6. Frequency analysis of first and second derivation operators (discrete case) 84

2.9. The two-dimensional discrete Fourier transform (2D DFT) 88

2.9.1. Matrix representation of the 2D DFT: Kronecker product versus iteration of two 1D DFTs 91

2.9.2. Properties of the 2D DFT 93

2.9.3. 2D DFT and stationary operators 95

2.9.4. Gradient and Laplace operators and their action on digital images 97

2.9.5. Visualization of the amplitude spectrum in 2D 97

2.9.6. Filtering: an example of digital image filtering in a Fourier space 100

2.10. Summary 102

Chapter 3. Lebesgue's Measure and Integration Theory 105

3.1. Riemann versus Lebesgue 105

3.2. -algebra, measurable space, measures and measured spaces 106

3.3. Measurable functions and almost-everywhere properties (a.e) 108

3.4. Integrable functions and Lebesgue integrals 109

3.5. Characterization of the Lebesgue measure on and sets with a null Lebesgue measure 111

3.6. Three theorems for limit operations in integration theory 113

3.7. Summary 114

Chapter 4. Banach Spaces and Hilbert Spaces 115

4.1. Metric topology of inner product spaces 116 4.2. Continuity of fundam...