For distributions the scaling property, $f(ax) = \frac{1}{|a|} \mathcal{F(\frac{u}{a})}$, of the Fourier transform is no longer true. Is there a source that lists which properties of the Fourier transform remain true even for distributions and which are false?

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    $\begingroup$ Are you sure about the failure of the scaling property? $\endgroup$ – Giuseppe Negro Dec 22 '12 at 1:30
  • $\begingroup$ I agree with Guiseppe. The scaling property is commonly exploited with Dirac delta functions. $\endgroup$ – Ron Gordon Dec 22 '12 at 4:23

A notable difference is behavior at $\infty$: The Fourier transform of a function in $L^1(\mathbb R)$ vanishes at $\infty$, but for a distribution it can remain constant (like the transform of $\delta$) or even grow (like the transform of $\delta'$).

But many properties can be passed from Schwartz functions $\varphi$ to distributions $f$, thanks to the definition $$\langle \widehat f, \varphi\rangle = \langle f, \widehat \varphi\rangle \tag1$$ For example, the scaling property passes from $\varphi$ to $f$. Stretching $f$ on the right of (1) has the same effect as shrinking $\widehat \varphi$ (and multiplication by some constant), which amounts to stretching $\varphi$, which has the same effect on the left as shrinking $\widehat f$.


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