# Faculty of Engineering and Science

The International Doctoral School of Technology and Science

#
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.
*L*^{2} 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.