MATH-205 / 7 crédits

Enseignant: Colombo Maria

Langue: Anglais

## Summary

Learn the basis of Lebesgue integration and Fourier analysis

## Content

Lebesgue integral

- Measurable sets and functions
- Lebesgue integral
- Monotone and dominated convergence theorems
- L^p spaces

Fourier analysis
- Fourier series
- Introduction to Fourier transform
- Applications to partial differential equations

## Required courses

Analysis I, II, III

## Learning Outcomes

By the end of the course, the student must be able to:

• Describe the fundamental concepts on the Lebesgue measure, the Lebesgue integration and the Fourier series/transform
• Define the objects and prove their properties
• Solve exercises and identify meaningful examples
• Use the Fourier series/transform to solve linear PDEs

## Teaching methods

Lectures and assisted/discussed exercises

## Assessment methods

• Written exam. A midterm will be organized and the final grade will be assigned according to a formula like
Final grade = \max { Final grade, 0.4 * Midterm grade + 0.6 * Final grade }

## Supervision

 Assistants Yes

## Bibliography

Polycopié of the course

T. Tao: "Analysis II"
B. Dacorogna: Polycopié

E. Stein: "Real analysis: measure theory, integration, and Hilbert spaces"
E. Stein: "Fourier analysis: an introduction"

S.D. Chatterji: "Cours d'analyse 1 et 3" PPUR
S.D. Chatterji: "Equations différentielles ordinaires et aux dérivées partielles"

## Prerequisite for

Master cycle of mathematics

## Dans les plans d'études

• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Analysis IV - Lebesgue measure, Fourier analysis
• Cours: 3 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Type: obligatoire

## Cours connexes

Résultats de graphsearch.epfl.ch.