# PhD Course

## Fourier Analysis

The official description of the course etc. can be found here.

### Description

A brief description of the plans at present. Will be updated as the course progresses.
February 1, 2005.
Introduction to the course. Background material on metric spaces, normed vector spaces, and inner product spaces. Orthonormal bases. Introduction to Fourier series. L2 theory for Fourier series. Derivation of the formula for the Dirichlet kernel. Further details and exercises are here. Note that these exercises are for you to practice on, to check your understanding of the material in the book.
February 7, 2005
The lectures today dealt with various aspects of pointwise convergence for Fourier series. Details and exercises are here. The first problem set is below, under the heading Evaluation.
February 14, 2005
I continued the discussion of the pointwise convergence of Fourier series, and summability methods. Details and exercises are here. The second problem set is below, under the heading Evaluation.
February 21, 2005
I presentd a number of results and computations on the continuous Fourier transform. Details and exercises are here. The third problem set will not be given. It turned out to be too complicated in relation to what I have covered concerning the continuous Fourier transform.
March 4, 2005
I presented results on the discrete Fourier transform, the discrete uncertainty principle, and its application to signal recovery.Details and exercises are here. There will be no fourth problem set.

### Course material

I use the book

G. Bachman, L. Narici, E. Beckenstein: Fourier and Wavelet Analysis, Springer-Verlag 2002.

I refer to it as BNB.

#### Supplementary material

Concerning the theory of distributions, mentioned in the lecture February 21, then I can recommend the book:

I. Richards, H. Youn: Theory of distributions: a non-technical introduction.
Cambridge University Press 1990

### Evaluation

Concerning evaluation of the participants, then I will assign 4 exercises, starting on February 7. To pass the course you should hand in solutions to these exercises.

#### Problem set 1

Here is the February 7 problem set. A written solution is to be handed in, no later than March 11, 2005.

#### Problem set 2

Here is the February 14 problem set. A written solution is to be handed in, no later than March 11, 2005.

As explained in a mail message, there will be no additional problem sets, thus 2 sets in all. Absolutely last date for submission is March 22, 2005.

Updated March 8, 2005,by Arne Jensen.