Faculty of Engineering and Science
The International Doctoral School of Technology and Science
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
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
- Section 5.9
- Concerns inversion, and was covered in detail.
- Section 5.10
- Concerns inversion in trigonometric form, and was not covered. Can
- 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.
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!
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
- Some examples of Fourier transforms
- Fourier transforms, derivatives and integrals
- Some complicated examples, showing that Maple may need assistance
Updated February 22, 2005, by Arne Jensen.