Integration

Objectives

The purpose of the course is to become acquainted with the Lebesgue integral and the use of the theory of integration, as it is involved in the tools for signal processing regarding integral transforms. The concept of distribution is also introduced as it is essential fo the generalization of basic operations as derivation, convolution or Fourier transform.

Description

Measure theory.
Measurable function, simple function integration.
Convergence theorems, Leibniz integral rule.
Lp spaces.
Distribution, derivation and convolution.
Fourier and Laplace transforms. 

Targeted skills

Be able to assess measurability and integrability of a given function.
Be able to apply convergence theorems and Leibniz integral rule.
Be able to use distributions, in particular in the resolution of Ordinary Differential Equations.
Be able to find Fourier and Laplace transforms, and to interpret them in the frame of signal processing.
 

Bibliography

Walter Rudin, Real and Complex Analysis, 3rd ed., Mc Graw Hill, 1987.
Laurent Schwartz, Méthodes mathématiques pour les sciences physiques, Hermann, 1965.
Walter Appel, Mathématiques pour la physique et les physiciens, 5ème ed., H&K, 2017.
Terence Tao, An introduction to measure theory, 2011.

Session 1 ou session unique - Contrôle des connaissances