# Harmonic Analysis

## Lecture hours

lecture

Monday, 10am in HeHo18, room 120
Thuesday, 8am in HeHo18, room 120

Freitag, 10am in HeHo18, room 120

## Content

"Harmonic analysis" is the branch of mathematics which is concerned with the representation of functions as a linear combination of elementary waves. The question of how to write a function as a superposition of waves is at least the origin of this mathematical theory and gave him its name. The main focus is therefore on Fourier series and Fourier transforms. One of our main interest is to find good criteria for Fourier multipliers. Fourier multipliers are sequences, which we can multiply every Fourier coefficients of some function space with and still get a Fourier coefficient of the same space. The dependency of the function space involved is obvious. For the space of square integrable functions the description of Fourier multipliers is easy. In contrast on the space of p-integrable functions a complete descriptions is not available yet. As Fourier multipliers on the space of p-integrable functions are of tremendous importance in analysis, sufficient conditions for these will play a major role in this lecture.

In this lecture we will also meet some modern developments. Indeed in the last 20 years theorems of harmonic analysis have emerged, which are extremely useful in other branches of mathematics such as partial differential equations and probability theory.

Our lecture will not only give you an introduction to this powerful theory but also give you basic training in analysis:

We provide an introduction to the theory of distributions and we will develop other techniques which will be handy in other parts of analysis (like convolution, integral kernel operators and especially interpolation theory).

The topics we are going to discuss:

• tempered distributions
• Fourier transform
• real interpolation
• Calderon Zygmund theory
• Littlewood Paley inequalities
• Fourier multiplier theorems
• applications for partial differential equations

## Literature

We recommend:

• Loukas Grafakos: Classical Fourier Analysis
• Loukas Grafakos: Modern Fourier Analysis

Other selected books about harmonic analysis:

• Elias Stein with Timothy S. Murphy: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals
• Yitzhak Katznelson: An introduction to harmonic analysis
• R.E. Ewards, G.I. Gaudry: Littlewood-Paley and Multiplier Theory.

A copy of each of these books (which are not available for borrowing and hence wait there for you the hole semester) can be found in the library (Semesterapperat).

## Prerequisites

• introductory lectures in analysis and linear algebra
• basic measure theory
• not required, but may be beneficial: a basic course in functional analysis

### Lecture Team

lecturer: Prof. Dr. Wolfgang Arendt

exercises: Manuel Bernhard

### Extent

4+2 hours per week, 9 ECTS

### Examination

There will be an oral exam at the end of the semester. As a permission to take the exam you have to reach 50% of the regular points in the exercise sheets.