From Fourier Analysis to Wavelets

✍ Scribed by Jonas Gomes, Luiz Velho

• Publisher: Springer International Publishing
• Year: 2015
• Tongue: English
• Leaves: 216
• Series: IMPA Monographs 3
• Edition: 1st ed. 2015
• Category: Library

✦ Synopsis

This text introduces the basic concepts of function spaces and operators, both from the continuous and discrete viewpoints. Fourier and Window Fourier Transforms are introduced and used as a guide to arrive at the concept of Wavelet transform. The fundamental aspects of multiresolution representation, and its importance to function discretization and to the construction of wavelets is also discussed.

Emphasis is given on ideas and intuition, avoiding the heavy computations which are usually involved in the study of wavelets.  Readers should have a basic knowledge of linear algebra, calculus, and some familiarity with complex analysis. Basic knowledge of signal and image processing is desirable.

This text originated from a set of notes in Portuguese that the authors wrote for a wavelet course on the Brazilian Mathematical Colloquium in 1997 at IMPA, Rio de Janeiro.

✦ Table of Contents

Front Matter....Pages i-xiii
Introduction....Pages 1-11
Function Representation and Reconstruction....Pages 13-27
The Fourier Transform....Pages 29-46
Windowed Fourier Transform....Pages 47-60
The Wavelet Transform....Pages 61-73
Multiresolution Representation....Pages 75-88
The Fast Wavelet Transform....Pages 89-100
Filter Banks and Multiresolution....Pages 101-112
Constructing Wavelets....Pages 113-125
Wavelet Design....Pages 127-141
Orthogonal Wavelets....Pages 143-155
Biorthogonal Wavelets....Pages 157-177
Directions and Guidelines....Pages 179-183
Back Matter....Pages 185-210

✦ Subjects

Fourier Analysis; Abstract Harmonic Analysis; Signal, Image and Speech Processing