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Fourier Analysis (Winter Semester 2014/15)


The results of the exam are now posted between the rooms 3A-05.1 and 3A-05.2. You may look at your exam on Friday, March 6, from 10:00 to 11:00 am in room 3A-27.

Lecture: Monday 11:30-13:00 1C-03
Monday 11:30-13:00 SR 2.58
Wednesday 14:00-15:30 1C-01
Wednesday 14:00-15:30 SR 2.58
Problem class: Tuesday 17:30-19:00 Z 2
Tuesday 17:30-19:00 SR 2.58
Lecturer Prof. Maria Girardi
Office hours: by appointment
Room 2.045 Kollegiengebäude Mathematik (20.30)
Email: girardi@math.sc.edu
Problem classes Dr. Martin Spitz
Office hours:
Room Kollegiengebäude Mathematik (20.30)


One should be familiar with basic concepts of Lebesgue Integration Theory (e.g. Hölder's inequality and Lebesgue's Dominated Convergence Theorem) as well as Hilbert and Banach spaces.

Exercise Sheets

Exercise Sheet 1
Exercise Sheet 2
Exercise Sheet 3
Exercise Sheet 4
Exercise Sheet 5
Exercise Sheet 6
Exercise Sheet 7
Exercise Sheet 8
Exercise Sheet 9
Exercise Sheet10
Solution Exercise Sheet 10



As needed and upon request, references will be added as the semester progresses. Here is a start.

  • Folland, Gerald B., Real analysis. Modern techniques and their applications. Second edition. John Wiley & Sons, Inc., New York, 1999. ISBN: 0-471-31716-0.
  • Werner, Dirk, Funktionalanalysis. Third edition. Springer-Verlag, Berlin, 2000. ISBN: 3-540-67645-7.

German/English Assistance

We do not want English to be a barrier in this class. If you do not understand the English, just let us know so that we can repeat and/or rephrase. Below are some resources.

  • Prof. Girardi's mathematical German-English cheat sheet: pdf and ods.


Reviews of some prequisities
Differentiation under the Integral
Lebesgue's Differentiation Theorem and Lebesgue Sets
Some Basics of Integration on R^N
Multi-index Review
Nets, cleaner version
Review of Topology
Urysohn's Lemma

Part I: Fourier Series
Tne Circle Group
Properties of the Fourier Coefficients
Table of Fourier Series

Part II: Fourier Transform
Convolution Examples
Fourier Transform Introduction
Families Of Seminorms Generating the Schwartz Class
Riesz-Thorin Interpolation Theorem
Proof of Riesz-Thorin Interpolation Theorem
Distributions Introduction