Wavelets: Mathematics and Applications (Studies in Advanced Mathematics)
Wavelets is a carefully organized and edited collection of extended survey papers addressing key topics in the mathematical foundations and applications of wavelet theory. The first part of the book is devoted to the fundamentals of wavelet analysis. The construction of wavelet bases and the fast computation of the wavelet transform in both continuous and discrete settings is covered. The theory of frames, dilation equations, and local Fourier bases are also presented.
The second part of the book discusses applications in signal analysis, while the third part covers operator analysis and partial differential equations. Each chapter in these sections provides an up-to-date introduction to such topics as sampling theory, probability and statistics, compression, numerical analysis, turbulence, operator theory, and harmonic analysis.
The book is ideal for a general scientific and engineering audience, yet it is mathematically precise. It will be an especially useful reference for harmonic analysts, partial differential equation researchers, signal processing engineers, numerical analysts, fluids researchers, and applied mathematicians.
Why Read This Book
You should read this book if you want a rigorous, research‑level tour of wavelet theory that connects deep mathematical foundations to concrete signal‑processing applications. You will learn how wavelet bases and frames are constructed, how fast wavelet transforms work in discrete and continuous settings, and how these tools are applied to compression, statistical analysis, and PDEs.
Who Will Benefit
Ideal for advanced graduate students, researchers, and engineers with strong mathematical background who need a precise, survey‑style reference on wavelet theory and its applications in signal processing and analysis.
Level: Advanced — Prerequisites: Undergraduate‑level real and complex analysis, linear algebra, Fourier analysis, and basic familiarity with digital signal processing and probability; some functional analysis or PDE background is helpful.
Key Takeaways
- Understand the mathematical foundations of wavelet analysis, including multiresolution analysis and dilation equations.
- Implement and analyze fast wavelet transform algorithms in both discrete and continuous frameworks.
- Apply frame theory and local Fourier bases to construct flexible signal representations.
- Design and evaluate wavelet‑based methods for compression, spectral analysis, and denoising.
- Use wavelet techniques in numerical analysis and operator/PDE contexts for solving and approximating differential problems.
- Assess statistical and probabilistic aspects of wavelet methods for signal and stochastic process analysis.
Topics Covered
- Introduction and historical overview of wavelets
- Foundations of multiresolution analysis
- Construction of orthonormal and biorthogonal wavelet bases
- Dilation equations and scaling functions
- Continuous wavelet transform and frame theory
- Discrete wavelet transform and fast algorithms
- Local Fourier bases and time‑frequency localization
- Applications to sampling, compression, and audio/speech analysis
- Statistical signal processing with wavelets (denoising, estimation)
- Wavelets in numerical analysis and approximation theory
- Operator analysis and applications to partial differential equations
- Open problems and directions in wavelet research
How It Compares
Complementary to Ingrid Daubechies' Ten Lectures on Wavelets (foundational and concise) and Stéphane Mallat's A Wavelet Tour of Signal Processing (more application‑oriented); this volume is broader as a collection of rigorous survey articles bridging pure math and applications.
