An Introduction to Wavelets Through Linear Algebra (Undergraduate Texts in Mathematics) by Michael W. Frazier (PDF)

Description

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.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐i use this book in order to write my bachelor thesis, it is great, very dynamic, and with a lot of examples and exercises.It also has an excellent walk through from linear algebra to wavelets, giving a more understanding point of view of this signal analysis tool.

⭐Wavelet theory is one of the newest branches of mathematics, originally emerged from the broader harmonic analysis, but now making its own way because of its multiple applications.There are several possible approaches to the subject, but maybe this one is both the easiest and the best one. Modern analysis relies more and more on operator theory (linear algebra in infinite-dimensional spaces) so this approach fits nicely in the overall framework of modern mathematics.This kind of introductory expositions are essential for a subject to get widespread, and this one really deserves full attention because by using these kind of techniques we are now able to solve better a lot of problems involving pulses, signals, oscillations, etc. Right, this is achieved normally using Fourier methods, but mathematics has improved a lot since the times of Fourier, and now we know that classical Fourier analysis is not always suitable for our particular needs. Wavelet analysis provides a means for constructing a specific Fourier-like method to solve our problem according to its intrinsic nature.Contents: Prologue: Compression of the FBI Fingerprint Files; Background: Complex Numbers and Linear Algebra; The Discrete Fourier Transform; Wavelets on Z_N; Wavelets on Z; Wavelets on R; Wavelets and Differential Equations.Originally intended for undergrads, but useful as a more advanced reference. Includes full explanations and lots of excercises. Extensive bibliography. Nice hardbound (as usual in Springer).

⭐Although the notation can be a little tight, the explanation of theconstruction of a basis using wavelets is very well developed. He startswith finite dimensional spaces and then moves onto wavelets on the realline. He has a good series of graphs of signals and their correspondingwavelets transforms and points out meaningful features of the wavelettransform graph.

⭐I consider this one of the best books on both Fourier and wavelet theory. The introduction on linear algebra is very clear and the presence of multiple exercises makes the book ideal for professors and students. Before passing to more complicated and complete books as, e.g., Mallat’s, I strongly suggest to study this one.

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