An Introduction to Wavelets Through Linear Algebra (Undergraduate Texts in Mathematics)
Wavelet theory is on the boundary between mathematics and engineering, making it ideal for demonstrating to students that mathematics research is thriving in the modern day. Students can see non-trivial mathematics ideas leading to natural and important applications, such as video compression and the numerical solution of differential equations. The only prerequisites assumed are a basic linear algebra background and a bit of analysis background. Intended to be as elementary an introduction to wavelet theory as possible, the text does not claim to be a thorough or authoritative reference on wavelet theory.
Why Read This Book
You will get a clear, linear-algebra-first introduction to wavelet theory that shows how concrete matrix and vector-space ideas produce orthonormal wavelet bases and fast transforms. The book emphasizes intuition and elementary proofs, so you can quickly connect abstract theory to practical constructions used in compression, numerical PDEs, and signal analysis.
Who Will Benefit
Undergraduate mathematics students, engineering students, and practicing engineers who want a concise, mathematically grounded introduction to wavelets without heavy measure-theoretic machinery.
Level: Beginner — Prerequisites: Basic linear algebra (inner product spaces, orthonormal bases, eigenvalues) and some familiarity with elementary analysis or Fourier series; no advanced measure theory required.
Key Takeaways
- Understand the multiresolution analysis framework and how it leads to wavelet bases
- Construct scaling functions and orthonormal wavelets using linear-algebraic techniques
- Relate discrete filter-bank operations to wavelet transforms and the fast wavelet algorithm
- Analyze examples such as the Haar and Daubechies wavelets and compute their basic properties
- Apply wavelet ideas to simple problems in compression and the numerical solution of differential equations
Topics Covered
- Introduction and motivation
- Review of linear algebra and L2 basics
- Fourier transforms and preliminaries
- Multiresolution analysis (MRA)
- Scaling functions and refinement equations
- Construction of orthonormal wavelets
- Examples: Haar, spline, and Daubechies-type wavelets
- Discrete wavelet transform and filter-bank interpretation
- Fast wavelet transform and algorithmic aspects
- Applications to approximation, compression, and PDEs
- Further topics and directions
- Appendices (background material)
How It Compares
Softer and more linear-algebra oriented than Daubechies' Ten Lectures (which is deeper/technical) and more mathematically focused and concise than Mallat's A Wavelet Tour of Signal Processing (which is broader and more application-driven).
