# PhD Course

## Fourier Analysis

### February 14, 2005

The lectures dealt with further aspects of pointwise convergence for Fourier series, in particular Cesaro summability. We also started on the transform view in Fourier Analysis. The material presented and its relation to the book BNB, and comments on what to read, follows below:
Sections 4.10 and 4.12
I stated the two main results, Theorem 4.10.1 and Theorem 4.12.1, and gave the main ideas of the proofs. I also went through the details of Example 4.10.2.
Section 4.13
Not part of the course, but quite interesting to read.
Section 4.14
I have gone through the general idea of a summability method, and discussed Cesaro summability. The general case has not been covered, and can be omitted.
Section 4.15
I have presented the main results in Fejer theory, and illustrated the convergence properties using Maple. The proofs have not been covered, and can be omitted.
Section 4.16
I went through this, using Maple to illustrate the smoothing properties.
Sections 4.17 and 4.18
Not covered, can be omitted.
Section 4.19 and 4.20
I presented an outline of the L2-theory in two variables, and discussed the problem in summing over two integer variables, something not covered in the book.
Section 5.1, 5.2, 5.3
These sections were covered in outline form. We will return to some of the results next time.

#### Problem set 2

The second problem set is here, as a .pdf file.

#### Exercises

The following exercises are suggested:
Section 4.10
Exercise 1
Section 4.14
Exercise 1
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.

Cesaro summability. some illustrations
cesaro.mw
Variants of the implementation of Cesaro summability
cesarovariants.mw
The Fejer and the Dirichlet kernel compared
fejer.mw
Multiple Fourier series, an example
multipleseries.mw

Updated February 16, 2005, by Arne Jensen.