Properties of the Fourier transform

Thursday, March 16. 2006

Properties of the Fourier transform

What I hate is when books say that something is clear, what definitely is not clear! So, why is it clear that $(\mathscr{F}\gamma_a)(\xi) = \frac{1}{a^n}(\mathscr{F}\gamma)\left(\frac{\xi}{a}\right)$ for $\gamma(x)=e^{-x^2/2}$ and $\gamma_a(x)=\gamma(ax)$ ? It seems to have something to do with the differentiation rules of the Fourier transform, and I couldn’t verify the mentioned equation yet. As far as I can see, it really is not that trivial that a simple clear-ness is justified.

This claim came together with $(\mathscr{F}\gamma)(\xi) = e^{-\xi^2/2}$ , the proof of which also doesn’t seem to be too trivial. What still confuses me is the symbolism $\textstyle x\xi=\sum_{i\in I}x_i\xi_i$ .

Posted by Stephan Paukner in Master's Thesis at 10:25 | Comments (0) | Trackback (1)

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Properties of the Fourier transform, II
I think I found out where the mentioned property comes from: The solution lies in the application of a substitution in the mentioned integral. It is well known that a translation of any (Lebesgue integrable) function leads to a modulation of its Fourier t

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