Wavelets: A Student Guide (Australian Mathematical Society Lecture Series, Series Number 24)
This text offers an excellent introduction to the mathematical theory of wavelets for senior undergraduate students. Despite the fact that this theory is intrinsically advanced, the author's elementary approach makes it accessible at the undergraduate level. Beginning with thorough accounts of inner product spaces and Hilbert spaces, the book then shifts its focus to wavelets specifically, starting with the Haar wavelet, broadening to wavelets in general, and culminating in the construction of the Daubechies wavelets. All of this is done using only elementary methods, bypassing the use of the Fourier integral transform. Arguments using the Fourier transform are introduced in the final chapter, and this less elementary approach is used to outline a second and quite different construction of the Daubechies wavelets. The main text of the book is supplemented by more than 200 exercises ranging in difficulty and complexity.
Why Read This Book
You should read this book if you want a clear, mathematically rigorous yet accessible introduction to wavelet theory that doesn’t assume advanced functional analysis. You will learn the core ideas behind multiresolution analysis, the Haar wavelet, and the celebrated Daubechies constructions through elementary arguments, with Fourier-based methods introduced later as an alternative viewpoint.
Who Will Benefit
Senior undergraduates or early graduate students in mathematics, electrical engineering, or signal processing who want a principled, proof-oriented entry to wavelets and multiresolution methods.
Level: Intermediate — Prerequisites: Calculus and linear algebra, basic real analysis (limits, sequences, and convergence); familiarity with inner-product concepts is helpful but the book reviews inner product spaces and Hilbert spaces first.
Key Takeaways
- Understand the structure of inner-product spaces and Hilbert spaces as the setting for wavelet theory
- Construct and analyze the Haar wavelet and general orthonormal wavelet bases using elementary methods
- Derive and construct compactly supported orthogonal wavelets, including Daubechies wavelets
- Apply multiresolution analysis to decompose L2(R) and relate wavelet bases to filter-bank interpretations
- Compare elementary (time-domain) constructions with Fourier-based approaches to wavelets
Topics Covered
- Preface and introduction to wavelets
- Inner product spaces and orthogonality
- Hilbert spaces and L2(R)
- Orthogonal bases, Riesz bases, and projections
- The Haar system and its properties
- Multiresolution analysis (MRA)
- Wavelets in L2(R) and constructions of wavelet bases
- Filter banks and the connection to discrete transforms
- Construction of Daubechies wavelets (compact support, orthogonality)
- Fast wavelet transform and discrete wavelet transform (discussion)
- Fourier-transform methods and an alternative construction
- Exercises, examples, and further reading
How It Compares
More elementary and undergraduate-oriented than Daubechies' Ten Lectures on Wavelets, and less application-heavy than Stéphane Mallat’s A Wavelet Tour of Signal Processing — Nickolas sits between rigorous theory and accessible pedagogy.
