# Faculty of Engineering and Science

The International Doctoral School of Technology and Science

#
PhD Course

## Fourier Analysis

### February 7, 2005

The lectures dealt with various aspects of pointwise convergence for
Fourier series. The material presented and its relation to the book
BNB, and comments on what to read follows below:
- Sections 4.1 and 4.2
- I reviewed the definitions and gave some examples of half-range
expansions (sine and cosine series).
- Section 4.3
- I illustrated the Weierstrass example of a `bad' function, using
Maple. The definitions of piecewise continuous functions, and the
smoothness concept are important, and should be studied in
detail. Omit the results on functions of bounded variation, pages 164-165.
- Section 4.4
- We only need the Riemann-Lebesgue Lemma in the form given in
equation (4.22). I gave a short argument for this, using
approximation by piecewise constant functions and integration by
parts. Omit the technical details and generalizations in this
section. They are not needed later.
- Section 4.5
- I have gone through the details of the derivation of the closed
form of the Dirichlet kernel, using the method in Exercise 4.5-1. I
have also presented in detail the results in 4.5.1, 4.5.3, and
4.5.4, and briefly mentioned the localization principle in
4.5.11. The other results in this section are optional, and give a
much more detailed understanding of aspects of pointwise
convergence, some of them rather technical.
- Section 4.6
- I have gone through pages 188-191, simplifying the argument by
directly rewriting the expression for the Dirichlet kernel and using
the Riemann-Lebesgue Lemma. The pages 192-194 can be omitted.
- Section 4.7
- I went through the details of this section.
- Section 4.8
- I went through the results on Gibbs phenomenon, and gave some
Maple illustrations.
- Section 4.9
- Omit this section.
- Section 4.10
- I presented the main result 4.10.1, without going into the
details.
- Section 4.11
- I discussed the difference between Fourier series and
trigonometric series, and gave Fatou's example. The example was
illustrated with Maple computations.

#### Problem set 1

The first problem set is here, as a .pdf file.
#### Exercises

The following exercises are suggested:
- Section 4.6
- Exercises 1 and 4.
- Section 4.10
- Go through the details of examples 4.10.2 and 4.10.3.

Note that there are hints for many of the exercises.
#### Maple files

The Maple worksheets that I used in the lectures are below. Note that
I have removed the output to reduce the size of the files. Thus you
should run them through Maple to see the results. This is most easily
done by clicking on the !!! in the toolbar.
Note that the worksheets are created using Maple 9.5. They are not
compatible with versions 8 or earlier.

- Examples of Fourier series
- feb7a.mw
- Further examples
- feb7b.mw
- Some `bad' functions
- feb7c.mw
- The Dirichlet kernel
- feb7d.mw
- Gibbs pehnomenon
- feb7e.mw
- Piecewise smooth functions
- feb7f.mw

Updated February 8, 2005, by Arne Jensen.