# PhD Course

## Fourier Analysis

### February 21, 2005

The lectures dealt with the continuous Fourier transform and its properties. I also presented the Heisenberg Uncertaintly Principle and the Donoho-Stark Uncertaintly Principle. I started on the discrete Fourier transform. The last lecture will be devoted mainly to the discrete Fourier transform. The material presented and its relation to the book BNB, and comments on what to read, follows below:
Sections 5.1, 5.2, 5.3, and 5.4
These sections contain motivation for the Fourier transform on the line. Optional reading.
Section 5.5
The basic definitions and properties of the Fourier transform. Covered in detail.
Section 5.6
The residue calculus to computing Fourier transforms was mentioned briefly. It is implemented in Maple, and I demonstrated a few computations using this approach. See the Maple files below.
Section 5.7
Mapping properties from L1 to C0 was presented in detail.
Section 5.8
Concerns convolution, and was presented in detail. A nice Maple animation explaining the smoothing property of convolution can be found here.
Section 5.9
Concerns inversion, and was covered in detail.
Section 5.10
Concerns inversion in trigonometric form, and was not covered. Can be omitted.
Sections 5.11 and 5.12
I gave an overview of the main result, using the condition that the original function to be recovered from its Fourier transform is piecewise continuous. You may omit the more refined results and the example showing that the Fourier map here is not onto.
Section 5.13
This section concerns general methods to make integrals converge better. I have only covered the first few pages. The many detail from page 319 onwards can be omitted.
Sections 5.14, 5.15, and 5.16
Omit these sections.
Section 5.17
Concerns Parseval identities, covered in detail.
Section 5.18
The L2 was discussed in some detail.
Sections 5.19-5.23
Not covered. Omit.
Uncertainty Principles
There will be some written material available on these topics later. You will receive information via mail.

#### Problem set 3

There will be no problem set 3 concerning the continuous Fourier transform.

#### Exercises

The following exercise is suggested:
Maple exercise
Using the worksheets below as a guide, try to compute the Fourier transform of a number of functions. Take examples from your own work, if relevant.
Maple exercise
Experiment with the worksheet explicit.mw below to see how rational functions are handled, or not handled, by Maple. Rational functions that have a Fourier transform are those that are quatients of two polynomials, with the degree of the denominator at least two larger that the degree of the numerator. Poles on the real axis may cause trouble. Try to experiment!

#### 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.

Overview of the transforms in the
```inttrans
```
package
overview.mw
Some examples of Fourier transforms
fourierex.mw
Fourier transforms, derivatives and integrals
deriv.mw
Some complicated examples, showing that Maple may need assistance factoring polynomials
explicit.mw

Updated February 22, 2005, by Arne Jensen.