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Institutes

Institute for Analysis

Workgroup Functional Analysis

Harmonic Analysis

Harmonic Analysis

Harmonic Analysis (Summer Semester 2021)

Lecturer: Prof. Dr. Dorothee Frey
Classes: Lecture (0156400), Problem class (0156410)
Weekly hours: 4+2

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: Tuesday, 10am - 11am, and 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.