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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 Meyer-Mallat 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

 

  1. Discrete Waves

  2. Continuous Waves

  3. Wavelet transform

  4. Frames

  5. Lipschitz regularity

  6. Compression applications