Vorlesung: Distributions and Operators (WiSe 2026-27)

!!!! TO BE UPDATED !!! PLEASE CHECK BACK !!!

Lecture (Vorlesung):

Thu 10-12 (in B 251).   LSF

Exercises (Übungen):
There are NO Exercises.

Synopsis (Kurzbeschreibung):
This course offers an introduction to distribution theory (generalized functions) by studying the special case of periodic distributions, that is, distributions on the 1d torus 𝕋 (or, circle 𝕊^1).
In the second part, we study various types of operators (translation, convolution, multiplication, integral, differential), in general, and concretely on 𝕋, with a view towards Functional Calculus (functions of operators) and applications to (partial) differential equations (PDEs).

(More details on content below.)

There will (maybe!) be a continuation (Pseudodifferential operators & PDE; 2 SWS) in the following semester.

NB Die Vorlesung wird auf Englisch gehalten.

Audience (Hörerkreis):
Master students of Mathematics (PO 2011: WP 17.2, 18.1, 18.2, 44.3, 45.2 or 45.3
- PO 2021: WP 13, 16, 40 or 41), TMP-Master.
Motivated Bachelor students of Mathematics are welcome;
they will get "Schein" if pass the course (contact the Lecturer).

Credits:
3 ECTS.

Prerequisites (Vorkenntnisse):
Analysis I-III, Linear Algebra I-II, and FA1.

Language (Sprache):
English. (Die mündliche Prüfung kan auch auf Deutsch gemacht werden).

Exam (Prüfung):
There will be an oral exam (30 min; dates to be announced) (Es wird eine mündliche Prüfung geben).
See separate webpage (Moodle).

Content (Inhalt) (!! TO BE UPDATED - CHECK BACK !!):

0. Introduction & Motivation
1. Testfunctions on T: smooth periodic functions
   1.1 Algebra & differentiation
   1.2 Translation, convolution & approximation
   1.3 Metric & continuity of operations
   1.4 Fourier series of smooth functions
   
2. Distributions on T: periodic distributions
2.1 Definition & examples
2.2 Algebra, translation & differentiation
2.3 Convolution & approximation
2.4 Fourier series of distributions
2.5 Support & structure of distributions
2.6 Sobolev spaces on T

3. Operators
3.1 Linear Algebra & The Spectral Theorem
3.2 Multiplication operators
3.3 Translation & convolution operators
3.4 Differential operators & Functional Calculus

4. Outlook: Pseudodifferential operators & PDE
4.1 Motivation: Functional Calculus
4.2 Symbols & operators
4.3 Linear evolution equations
4.4 Nonlinear evolution equations

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.

Supplementary literatur (Ergänzende Literatur) (!! TO BE UPDATED - CHECK BACK !!):

• [B] R. Beals, Advanced Mathematical Analysis, Springer GTM, 1973.
• [G] G. Grubb, Distributions and operators, Springer GTM, 2009.
• [HK] K. Hoffman, R. Kunze, Linear Algebra, Second Edition, Prentice-Hall, 1971.
• [I] R. Iorio, V. de M. Iorio, Fourier Analysis and Partial Differential Equations, CUP, 2001.
• [K1] W. Kaballo, Grundkurs Funktionalanalysis, 2. Auflage, Springer Spektrum, 2018.
• [K2] W. Kaballo, Aufbaukurs Funktionalanalysis und Operatortheorie, Springer Spektrum, 2014.
• [W1] J. Weidmann, Lineare Operatoren in Hilberträumen. Teil 1: Grundlagen, Teubner, 2000.
• [W2] D. Werner, Funktionalanalysis, Springer, 2018.

Office hours (Sprechstunde):
See Moodle.

To access the course material, you need to sign up (OPENS 5 Oct 2026) in Moodle here (Psword: Schwartz) .

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Letzte Änderung: 03 October 2026.

Thomas Østergaard Sørensen