Vorlesung: Fourier Series (WiSe 2025-26)
!!!! TO BE UPDATED !!! PLEASE CHECK BACK !!!
Lecture (Vorlesung):
Wed 10-12 (in B 004) & Fri 08-10 (in B 006). LSF
Exercises (Übungen):
See separate webpage (Moodle). LSF
Tutorials (Tutorien):
See separate webpage (Moodle).
Synopsis (Kurzbeschreibung):
Description to come!
(More details on content below.)
NB Die Vorlesung wird auf Englisch gehalten.
Audience (Hörerkreis):
Students pursuing the following degrees: BSc (Bachelor) Mathematics
(PO 2021: WP4 OR WP5; PO 2015: WP20).
Students in other study programmes should get in touch with the Lecturer.
Credits:
9 (6+3) ECTS (4+2 SWS).
Prerequisites (Vorkenntnisse):
Analysis I-III, Lineare Algebra I-II. Functional Analysis (FA) is an advantage
but not indispensable: motivated (!) students without FA are very welcome!
We will cover a lot of ground, relatively fast.
Language (Sprache):
English.
Exam (Prüfung):
See Moodle.
Content (Inhalt) (!! TO BE UPDATED - CHECK BACK !!):
-
Fourier analysis: Fourier coefficients; mapping properties of
the Fourier transform ℱ: L^1[0,2pi] -> l^infinity
(boundedness on various spaces; injectivity/surjectivity/what is ran(ℱ)?); Hölder & Sobolev spaces
(& their properties & embeddings) & mapping properties of ℱ; regularity
(C^k, C^infinity, analyticity, Hölder & Sobolev) & decay of Fourier coefficients. -
Fourier synthesis: Convergence of Fourier series (pointwise; uniformly (on subsets);
absolute; normal; pointwise a.e.; L^p-convergence; in measure; summability/regularization (Fejer/Cesaro, Abel, etc)
& convergence (in L^p & a.e.)); divergence of Fourier series (examples & counter examples);
connection to Fourier analysis. -
Applications: To PDE's: Wave, Heat & Laplace equation: separation of variables (convergence!);
eigenfunction expansions; eigenvalue asymptotics; Sturm-Liouville problems; Approximation Theory. - Function spaces: BV, absolutely continuous functions; Hölder & Sobolev spaces (properties & embeddings).
-
Operators: Convolution operators; integral operators (incl singular integral operators (SIO); Hilbert transform);
"good kernels"/approximation of the identity/Dirac sequences; maximal functions; interpolation. -
Various Outlooks (including generalisations of all (!) the above): Multi-dimensional Fourier series;
Fourier transform on R^d; Abstract Harmonic Analysis.
Literature:
In Moodle you will find a copy of the notes from the lecture (to be updated as we go along).
Above you will find a short description of the content of the lecture.
The course will not follow a particular textbook. The list below provides a short selection
of relevant English and German textbooks on the subject (of which there are many!).
Supplementary literatur (Ergänzende Literatur):
- A. Deitmar, A First Course in Harmonic Analysis, Springer, 2005.
- G. B. Folland, Fourier analysis and its application, AMS, 1992.
- J. B. Garnett, Bounded Analytic Functions, Academic Press, 1981.
- W. Kaballo, Grundkurs Funktionalanalysis, 2. Auflage, Springer Spektrum, 2018.
- W. Kaballo, Aufbaukurs Funktionalanalysis und Operatortheorie, Springer Spektrum, 2014.
- Y. Katznelson, An introduction to Harmonic Analysis, 3rd edition, CUP, 2004.
- M. A. Pinsky, Introduction to Fourier Analysis and Wavelets, AMS, 2009.
Office hours (Sprechstunde):
See Moodle.
To access the course material, you need to sign up (opens 06. October 2025) in Moodle here (Psword: Carleson) .
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Letzte Änderung: 28 July 2025.
Thomas Østergaard Sørensen