The International Doctoral School of Technology and Science

- 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
*L*to^{1}*C*was presented in detail._{0} - 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
*L*was discussed in some detail.^{2} - 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.

- 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!

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.