An Introduction to Wavelets (Volume 1) (Wavelet Analysis and Its Applications, Volume 1)
An Introduction to Wavelets is the first volume in a new series, WAVELET ANALYSIS AND ITS APPLICATIONS. This is an introductory treatise on wavelet analysis, with an emphasis on spline wavelets and time-frequency analysis. Among the basic topics covered in this book are time-frequency localization, integral wavelet transforms, dyadic wavelets, frames, spline-wavelets, orthonormal wavelet bases, and wavelet packets. In addition, the author presents a unified treatment of nonorthogonal, semiorthogonal, and orthogonal wavelets. This monograph is self-contained, the only prerequisite being a basic knowledge of function theory and real analysis. It is suitable as a textbook for a beginning course on wavelet analysis and is directed toward both mathematicians and engineers who wish to learn about the subject. Specialists may use this volume as a valuable supplementary reading to the vast literature that has already emerged in this field.
Key Features
* This is an introductory treatise on wavelet analysis, with an emphasis on spline-wavelets and time-frequency analysis
* This monograph is self-contained, the only prerequisite being a basic knowledge of function theory and real analysis
* Suitable as a textbook for a beginning course on wavelet analysis
Why Read This Book
You should read this book if you want a mathematically solid yet application-aware introduction to wavelet theory that explains spline wavelets, frames, and wavelet packets in depth. You will gain a unified view of orthogonal, semiorthogonal and nonorthogonal constructions and learn the theoretical foundations that underpin practical DSP uses of wavelets.
Who Will Benefit
Graduate students, DSP engineers, and researchers who need a rigorous grounding in wavelet theory—especially those working on spectral/time-frequency analysis, filter-bank design, or wavelet-based signal processing.
Level: Advanced — Prerequisites: Comfort with undergraduate real analysis and function theory, linear algebra, and basic Fourier analysis (signals and systems familiarity recommended).
Key Takeaways
- Explain the principles of time-frequency localization and its relation to wavelet analysis.
- Describe and contrast continuous and discrete (dyadic) wavelet transforms and their mathematical foundations.
- Construct and analyze spline-based wavelets and understand their approximation and regularity properties.
- Apply the theory of frames, Riesz bases, and multiresolution analysis to design and evaluate wavelet bases.
- Use wavelet packets and related decompositions for flexible time-frequency tiling and spectral analysis.
- Differentiate between orthogonal, semiorthogonal, and nonorthogonal wavelet systems and their implications for implementation.
Topics Covered
- Introduction and historical overview
- Mathematical preliminaries (function spaces, Fourier transform, distributions)
- Time-frequency localization and uncertainty principles
- Integral (continuous) wavelet transforms
- Dyadic wavelets and multiresolution analysis
- Construction and properties of spline wavelets
- Orthonormal, semiorthogonal and biorthogonal wavelet bases
- Frames, Riesz bases, and stability of expansions
- Wavelet packets and adaptive tilings
- Filter-bank realizations and discretization issues
- Selected applications and examples
- Appendices: technical lemmas and proofs
How It Compares
Compared with Daubechies' Ten Lectures on Wavelets (compactly supported orthonormal wavelets) this book emphasizes spline wavelets and a unified treatment of nonorthogonal systems; compared to Mallat's A Wavelet Tour, Chui is more mathematically rigorous and less implementation-focused.
