# Faculty of Engineering and Science

The International Doctoral School of Technology and Science

#
PhD Course

## Fourier Analysis

### February 1, 2005

I gave an introduction to basic concepts concerning metric spaces,
convergence, normed vector spaces and inner product
spaces. Orthonormal bases were introduced.
The basics of Fourier series were presented, up to the
*L*^{2} theory. The formula for the Dirichlet kernel was derived.

#### The book

The book BNB contains a lot of material. The chapters 1-3 contain
background material. The core text begins in chapter 4. Comments on
the individual sections in chapter 4 and the lectures, as far as I
have come.
- Section 4.1
- I assume that the material in this section is well known.
- Section 4.2
- Read through and understand the results.
- Section 4.3
- Pages 159-162: This is background material. I will illustrate
some of the phenomena next time. Pages 163-165: I will go through
these results next time.
- Section 4.4
- I will go through these results on Feb. 7.
- Section 4.5
- I have gone through the derivation of the formula for the
Dirichlet kernel, not as in the text, but as suggested in
Exercise 4.5-1. I will continue with this section on Feb. 7.

#### Exercises

The following exercises are suggested:
- Section 4.1
- Exercises 2 and 3.
- Section 4.2
- Exercise 1.
- Section 4.5
- Exercise 1. Draw the graph of the Dirichlet kernel for a number
of values of N.

Note that there are hints for many of the exercises.

Updated February 1, 2005,by Arne Jensen.