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):
There are NO Tutorials.

Synopsis (Kurzbeschreibung):
Fourier series is a classical topic in Analysis, (normally) presented in the Grundausbildung (Analysis I-III). In fact, historically, many topics presented in courses in the Bachelor studies (for example, convergence; the concept of a function (Dirichlet); the Riemann integral (Riemann), naive set theory (Cantor), the real numbers (Dedekind; Cantor; ..), uniform convergence (Weierstrass); ... ) were originally invented (or, found, or properly defined) to study and answer questions about Fourier series.
In this course we study more advanced questions (and their answers!) about Fourier series. We will also make certain connections between single- and multivariable Calculus (Ana1-3), Complex Analysis (Fkt WP17), Ordinary Differential Equations (ODE WP15), Functional Analysis (FAn WP19), and Partial Differential Equations (PDE WP23) more clear.
This can both be seen as application of certain results from these courses (if you have taken them), or, as a preliminary motivation for taking them later.
No previous knowledge of Complex Analysis, ODE, FA or PDE are needed (but is advantageous); we will simply use certain results and facts. References and Handouts will be provided.
On the other hand, the main aim of the course is to present more advanced questions and topics - topics that (may) be met in Analysis courses in the Master studies, and whose generalizations are often still at the forefront of current research in Analysis. This will illustrate how mathematical research works in practice.

(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/smoothness
  (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 (Fejér/Cesàro, Abel, Tauberian theorems 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;
  eigenfunction expansions (convergence!); eigenvalue asymptotics; Sturm-Liouville problems; Approximation Theory.
• Function spaces: BV, absolutely continuous functions; Hölder & Sobolev spaces (properties & embeddings).
• Operators: Convolution operators; multiplier operators; integral operators (incl singular integral operators (SIOs); Hilbert transform);
  summability kernels/"good kernels"/approximation of the identity/Dirac delta sequences; maximal functions; interpolation.
• Various Outlooks (including generalisations of all (!) the above): Carleson-Hunt Theorem, Multi-dimensional Fourier series;
  Fourier transform on R^d and Harmonic Analysis; 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) (!! TO BE UPDATED - CHECK BACK !!):

• [A] J. Arias de Reyna, Pointwise Convergence of Fourier Series, LNM 1785, Springer, 2002.
• [De] A. Deitmar, A First Course in Harmonic Analysis, Springer, 2005.
• [Du] J. Duoandikoetxea, Fourier Analysis, AMS, 2001.
• [E] R. E. Edwards, Fourier Series, Volume 1, 2nd Edition, Springer, 1979.
• [F] G. B. Folland, Fourier analysis and its application, AMS, 1992.
• [G] J. B. Garnett, Bounded Analytic Functions, Academic Press, 1981.
• [He] H. Helson, Harmonic Analysis, 2nd edition, Hindustan Book Agency, 2010.
• [Ho] K. Hoffman, Banach Spaces Of Analytic Functions, Prentice Hall Inc., 1962.
• [K-FA1] W. Kaballo, Grundkurs Funktionalanalysis, 2. Auflage, Springer Spektrum, 2018.
• [K-FA2] W. Kaballo, Aufbaukurs Funktionalanalysis und Operatortheorie, Springer Spektrum, 2014.
• [K] Y. Katznelson, An introduction to Harmonic Analysis, 3rd edition, CUP, 2004.
• [MS] C. Muscalu, W. Schlag, Classical and Multilinear Harmonic Analysis, Volume 1, CUP, 2013.
• [P] M. A. Pinsky, Introduction to Fourier Analysis and Wavelets, AMS, 2009.
• [Z] A. Zygmund, Trigonometric series, 3rd edition, CUP, 2003.

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: 07 October 2025.

Thomas Østergaard Sørensen