# Faculty of Engineering and Science

The International Doctoral School of Technology and Science

#
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
*L*^{2}-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.