Ten Lectures on Wavelets (CBMS-NSF Regional Conference Series in Applied Mathematics, Series Number 61)
This monograph contains 10 lectures presented by Dr. Daubechies as the principal speaker at the 1990 CBMS-NSF Conference on Wavelets and Applications. Wavelets are a mathematical development that many experts think may revolutionize the world of information storage and retrieval. They are a fairly simple mathematical tool now being applied to the compression of data, such as fingerprints, weather satellite photographs, and medical x-rays - that were previously thought to be impossible to condense without losing crucial details. The opening chapter provides an overview of the main problems presented in the book. Following chapters discuss the theoretical and practical aspects of wavelet theory, including wavelet transforms, orthonormal bases of wavelets, and characterization of functional spaces by means of wavelets. The last chapter presents several topics under active research, as multidimensional wavelets, wavelet packet bases, and a construction of wavelets tailored to decompose functions defined in a finite interval.
Why Read This Book
You should read this book if you want a compact, authoritative introduction to the mathematical foundations behind wavelets and the fast wavelet transform; it explains how compactly supported orthonormal wavelets are constructed and why they work. The lectures balance rigour and intuition so you will come away able to connect theory (multiresolution analysis, vanishing moments, regularity) to practical DSP uses like compression and signal approximation.
Who Will Benefit
Graduate students, DSP engineers, and researchers who need a rigorous foundation in wavelet theory to design or analyze wavelet-based algorithms for signal, image, and data compression or approximation.
Level: Advanced — Prerequisites: Linear algebra, basic Fourier analysis and signal processing concepts, and familiarity with undergraduate real analysis; some exposure to functional analysis or distribution theory is helpful but not required.
Key Takeaways
- Explain the concept of multiresolution analysis and how it underpins wavelet constructions.
- Construct compactly supported orthonormal wavelets (Daubechies wavelets) from scaling filters.
- Analyze vanishing moments and regularity properties and their impact on approximation and compression.
- Implement and reason about the fast wavelet transform (filter-bank viewpoint) used in practical DSP.
- Characterize function spaces (e.g., Besov spaces) using wavelet coefficients for approximation theory.
- Relate wavelet theory to practical applications such as data compression and sparse signal representation.
Topics Covered
- Lecture 1 — Overview and motivation: wavelets in signal processing and approximation
- Lecture 2 — Multiresolution analysis (MRA): scaling functions and nested spaces
- Lecture 3 — From filters to wavelets: refinement equations and filter-bank formulation
- Lecture 4 — Existence and construction of compactly supported orthonormal wavelets
- Lecture 5 — Daubechies' wavelets: vanishing moments and minimal support construction
- Lecture 6 — Regularity and smoothness of wavelets; Hölder and Sobolev properties
- Lecture 7 — Fast wavelet transform and implementation via filter banks
- Lecture 8 — Biorthogonal wavelets and spline-based constructions (overview)
- Lecture 9 — Wavelet characterization of function spaces (Besov and Triebel–Lizorkin)
- Lecture 10 — Applications and further directions: compression, numerical analysis, and open problems
How It Compares
More mathematically focused and compact than Stéphane Mallat's 'A Wavelet Tour of Signal Processing' (Mallat is broader and more applied), and more theoretical than Strang & Nguyen's 'Wavelets and Filter Banks', which emphasizes linear algebra and implementation.
