Harmonic Analysis (Winter Semester 2024/25)
- Lecturer: Prof. Dr. Dorothee Frey
- Classes: Lecture (0156400), Problem class (0156410)
- Weekly hours: 4+2
All information on the lecture is provided on the Ilias page on Harmonic Analysis.
| Schedule | ||
|---|---|---|
| Lecture: | Tuesday 8:00-9:30 | 20.30 SR 2.66 |
| Thursday 9:45-11:15 | 20.30 SR 2.66 | |
| Problem class: | Wednesday 15:45-17:15 | 20.30 - 01.25 |
| Lecturers | ||
|---|---|---|
| Lecturer | Prof. Dr. Dorothee Frey | |
| Office hours: Tuesday, 10am - 11am, and by appointment | ||
| Room 2.042 Kollegiengebäude Mathematik (20.30) | ||
| Email: dorothee.frey@kit.edu | Problem classes | M.Sc. Yonas Mesfun |
| Office hours: | ||
| Room 2.045 Kollegiengebäude Mathematik (20.30) | ||
| Email: yonas.mesfun@kit.edu | ||
This course gives an introduction to Fourier analysis and real harmonic analysis. Harmonic analysis has its origin in Fourier's work, which provides explicit solutions for e.g. heat and wave equations. Nowadays, it provides important tools in tackling a large variety of problems coming from partial differential equation, in particular for boundary value problems on L^p spaces. Topics of the course include summability of Fourier series, the Fourier transform on R^d, interpolation methods, singular integral operators, Fourier multipliers and oscillatory integrals.
Contents:
- Fourier series
- Fourier transform on L^1 and L^2
- Tempered distributions and Fourier transforms
- Explicit solutions of heat equation, wave equation and Schrödinger equation on R^d
- Hilbert transform
- Interpolation theorem of Marcinkiewicz
- Singular integral operators
- Mihlin multiplier theorem
- Oscillatory integrals
Examination
Oral exam of approx. 30 minutes (in English or German).
References
- Y. Katznelson. An introduction to harmonic analysis. Third edition. Cambridge University Press, Cambridge, 2004.
- L. Grafakos. Classical Fourier analysis. Third edition. Graduate Texts in Mathematics, 249. Springer, New York, 2014.
- E. M. Stein. Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970
- E. M. Stein; R. Shakarchi. Fourier analysis. An introduction. Princeton Lectures in Analysis, 1. Princeton University Press, Princeton, NJ, 2003.
- P. Auscher. Real Harmonic Analysis. ANU Press, 2012. https://press.anu.edu.au/publications/real-harmonic-analysis
- L. C. Evans; R. F. Gariepy. Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, 1992.