Webrelaunch 2020

Harmonic Analysis (Summer Semester 2021)

The course will be held online (at least at the beginning).

All relevant course information can be found on the study platform Ilias.

Schedule
Lecture: Monday 10:00-23:59 20.30 SR 3.61 Begin: 12.4.2021
Wednesday 8:00-23:59 20.30 SR 3.68
Problem class: Friday 10:00-11:30 20.30 - 01.25 Begin: 16.4.2021
Lecturers
Lecturer Prof. Dr. Dorothee Frey
Office hours: By appointment
Room 2.042 Kollegiengebäude Mathematik (20.30)
Email: dorothee.frey@kit.edu
Problem classes Dr. Nick Lindemulder
Office hours:
Room 2.044 Kollegiengebäude Mathematik (20.30)
Email: nick.lindemulder@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 \mathbb{R}^n, interpolation methods, singular integral operators, and Fourier multipliers.

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 \mathbb{R}^n
  • Hilbert transform
  • Interpolation theorem of Marcinkiewicz
  • Singular integral operators
  • Mihlin multiplier theorem


Recommendations:
Basic knowledge in Functional Analysis is recommended.

Examination

Oral exam of approx. 30 minutes.

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.