MATH-205 / 7 credits

Teacher: Colombo Maria

Language: English


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

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
  • Lecture: 3 Hour(s) per week x 14 weeks
  • Exercises: 2 Hour(s) per week x 14 weeks
  • Type: mandatory

Reference week

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