Fourier analysis and Wavelets
Fourier's analysis is one of the great achievements of mathematics, his ideas have profoundly influenced almost all areas of mathematics and physics. In addition to the above, it is important to highlight the large number of applications in engineering, signal theory and recently in data science. Recently these techniques have been generalized by means of wavelets for example for file compression. This course seeks to invite the student to learn the details behind these fascinating methods.
Objectives

A review of linear algebra with a view to Fourier analysis and the study of Wavelets.

Study the applications of the Fourier transform and the Haar bases, which represent the first steps for the Wavelet theory.

Introduce the student to the study of wavelets from a formal point of view in the MeyerMallat sense.

Know some applications of wavelets to reverse problems, compression or denoising.
Syllabus
Course one
1. Fourier analysis

Basic concepts

The discrete Fourier transform

The Fourier transform continues

Study of the Fourier transform in L1

Fundamental theorems

The fast Fourier transform

Compression applications

Heisenberg principle

Gibbs effect

Relationship with information theory
2. Gabor Transforms
3. Haar: the first wavelets

Haar Bases

Kronecker Product

Compression applications

Haar transform
Course two

Discrete Waves

Continuous Waves

Wavelet transform

Frames

Lipschitz regularity

Compression applications