Fourier Series and Wavelets

Title page
Preface
PART 1. FOURIER SERIES
Introduction. What are Fourier series about?
Chapter 1. Who was Fourier?
Chapter 2. The beginning of Fourier series
1. The analytical theory of heat. Introduction
2. Chapters I to III
3. Chapters IV to IX
4. Back to the introduction
Chapter 3. Predecessors and challengers
1. The prehistory of harmonic analysis
2. Vibrating strings, D. Bernoulli, Euler, and d'Alembert
3. Lagrange
4. Euler and Fourier formulas, Clairaut
5. Poisson and Cauchy
6. For further information
Chapter 4. Dirichlet and the convergence problem
1. Dirichlet
2. Comments on the article
3. The convergence problem since then
4. Dirichlet and Jordan
5. Dirichlet's original paper
6. A quotation of Jacobi
Chapter 5. Riemann and real analysis
1. Riemann
2. The memoir on trigonometric series. The historical part
3. The memoir on trigonometric series. The notion of integral
4. The memoir on trigonometric series. Functions representable by such series
5. The memoir on trigonometric series. The final section
6. Other special trigonometric series. Riemann and Weierstrass
7. An overview on the influence of Riemann's memoir just after 1867
8. A partial view on the influence of Riemann's memoir in the twentieth century
9. An excerpt from Riemann's memoir
Chapter 6. Cantor and set theory
1. Cantor
2. Cantor's works on trigonometric series
3. Über die Ausdehnung
4. Sets of uniqueness and sets of multiplicity
5. Two methods for thin sets in Fourier analysis
6. Baire's method
7. Randomization
8. Another look on Baire's theory
9. Recent results and new methods from general set theory
10. The first paper in the theory of sets
Chapter 7. The turn of the century and Fejér's theorem
1. Trigonometric series as a disreputable subject
2. The circumstances of Fejér's theorem
3. A few applications and continuations of Fejér's theorem
Chapter 8. Lebesgue and functional analysis
1. Lebesgue
2. Lebesgue and Fatou on trigonometric series (1902-1906)
3. Trigonometric series and the Lebesgue integral
4. Fatou-Parseval and Riesz-Fischer
5. Riesz-Fischer and the beginning of Hilbert spaces
6. L^p, l^q, functions and coefficients
7. L^p, H^p, conjugate functions
8. Functionals
9. Approximation
Chapter 9. Lacunarity and randomness
1. A brief history
2. Rademacher, Steinhaus and Gaussian series
3. Hadamard series, Riesz products and Sidon sets
4. Random trigonometric series
5. Application of random methods to Sidon sets
6. Lacunary orthogonal series. Λ(s) sets
7. Local and global properties of random trigonometric series
8. Local and global properties of lacunary trigonometric series
9. Local and global properties of Hadamard trigonometric series
Chapter 10. Algebraic structures
1. An inheritance from Norbert Wiener
2. Compact Abelian groups
3. The Wiener-Lévy theorem
4. The converse of Wiener-Lévy's theorem
5. A problem on spectral synthesis with a negative solution
6. Another negative result on spectral synthesis
7. Homomorphisms of algebras A(G)
Chapter 11. Martingales and H^p spaces
1. Taylor series, Walsh series, and martingales
2. A typical use of Walsh expansions: a best possible Khintchin inequality
3. Walsh series and dyadic martingales
4. The Paley theorem on Walsh series
5. The H^p spaces of dyadic martingales
6. The classical H^p spaces and Brownian motion
Chapter 12. A few classical applications
1. Back to Fourier
2. The three typical PDEs
3. Two extremal problems on curves
4. The Poisson formula and the Shannon sampling
5. Fast Fourier transform
References
Index
PART II. WAVELETS
Chapter 0. Wavelets: A brief historical account
1. Jean Morlet and the beginning of wavelet theory (1982)
2. Alex Grossmann and the Marseille team (1984)
3. Yves Meyer and the triumph of harmonic analysis (1985)
4. Stéphane Mallat and the fast wavelet transform (1986)
5. Ingrid Daubechies and the FIR filters (1987)
Chapter 1. The notion of wavelet representation
1. Time-frequency localization and Heisenberg's inequality
2. Almost orthogonal families, frames and bases in a Hilbert space
3. Fourier windows, Gabor wavelets and the Balian-Low theorem
4. Morlet wavelets
5. Wavelet analysis of global regularity
6. Wavelet analysis of pointwise regularity
Chapter 2. Discrete wavelet transforms
1. Sampling theorems for the Morlet wavelet representation
2. The vaguelettes lemma and related results for H_{ε,ε'} spaces
3. Proof of the regular sampling theorem
4. Proof of the irregular sampling theorem
5. Some remarks on dual frames
6. Wavelet theory and modern Littlewood-Paley theory
Chapter 3. The structure of a wavelet basis
1. General properties of shift-invariant spaces
2. The structure of a wavelet basis
3. Definition and examples of multi-resolution analysis
4. Non-existence of regular wavelets for the Hardy space H_{(2)}
Chapter 4. The theory of scaling filters
1. Multi-resolution analysis, scaling functions and scaling filters
2. Properties of the scaling filters
3. Derivatives and primitives of a regular scaling function
4. Compactly supported scaling functions
Chapter 5. Daubechies' functions and other examples of scaling functions
1. Interpolating scaling functions
2. Orthogonal multi-resolution analyses
3. Spline functions: the case of orthogonal spline wavelets
4. Bi-orthogonal spline wavelets
Chapter 6. Wavelets and functional analysis
1. Bi-orthogonal wavelets and functional analysis
2. Wavelets and Lebesgue spaces
3. H¹ and BMO
4. Weighted Lebesgue spaces
5. Besov spaces
6. Local analysis
Chapter 7. Multivariate wavelets
1. Multivariate wavelets: a general description
2. Existence of multivariate wavelets
3. Properties of multivariate wavelets
Chapter 8. Algorithms
1. The continuous wavelet transform
2. Mallat's algorithm
3. Wavelets on the interval
4. Quadrature formulas
5. The BCR algorithm
6. The wavelet shrinkage
Chapter 9. Further extensions of wavelet theory
1. Multiple scaling functions
2. Wavelet packets
3. Local sine bases
4. The matching pursuit algorithm
Chapter 10. Some examples of applications of wavelets to analysis
1. Wavelets and para-products
2. The div-curl theorem
3. Calderon-Zygmund operators
4. The Riemann function
References
Index