Wavelet Theory — where do I start?

I am in the process of implementing a Fixed-Point Fast Fourier Transform. The Fixed-Point FFT requires mathematical background in the area of wavelets and lifting schemes.

What are good resources/pre-requisites in order to learn wavelet theory? I have dived head first onto lifting schemes, but quickly realized that I am lacking in the mathematical background.

My math background goes all the way to Differential Equations to Discrete Math with some proofs. Would like to get a perspective on what I should know first before I do wavelet theory.

Thanks!

• Take a look at Ten Lectures on Wavelets by Daubechies. You will probably want to study functional analysis first, up to Hilbert spaces and their frames. A lot of advanced wavelet theory is about tweaking frames to achieve some desirable effect. – Alexei Averchenko Jul 4 '11 at 5:06
• Also, AFAIK working with wavelets on manifolds other than $\mathbb{R}^n$ requires some harmonic analysis, so if you want to do those, you may want to check it out. – Alexei Averchenko Jul 4 '11 at 5:15
• Thanks! I'll take a look at it. – Carlo del Mundo Jul 4 '11 at 15:16

Wavelet theory has developed as an interdisciplinary topic, so you'll find literature that explains the subject from the viewpoint of engineers and programmers, mathematicians with a background in discrete math, signal processing, system theory, or functional analysis.

One topic that everybody has to understand is basic Hilbert space theory (Hilbert spaces themselves and linear operators on them). There are a lot of books that cover that.

If your background is in signal processing, I'd recommend

• Stéphane Mallat: "A Wavelet Tour of Signal Processing, The Sparse Way", 3rd edition.

This book contains a lot of applications. An introduction that has a stronger focus on the mathematical background is this:

• Eugenio Herandez, Guido Weiss: "A First Course on Wavelets" (CRC Press 1996)

(BTW: Did you mean fixed-point FFT algorithms?).

• Yes, I meant fixed point. I'll give the Mallat book a shot! – Carlo del Mundo Jul 4 '11 at 15:15