Analysis IV - Lebesgue measure, Fourier analysis
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
Learning Prerequisites
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 |
Resources
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"
Ressources en bibliothèque
- Analysis II / Tao
- Polycopié / Dacorogna
- Real analysis / Stein
- Fourier analysis / Stein
- Cours d'Analyse (volume 1) / Chatterji
- Equations différentielles ordinaires et aux dérivées partielles / Chatterji
Moodle Link
Prerequisite for
Master cycle of mathematics
In the programs
- Semester: Spring
- Exam form: Written (summer session)
- Subject examined: Analysis IV - Lebesgue measure, Fourier analysis
- Courses: 3 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: mandatory