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I am really stuck at problems relating to this subject. This leads me to believe that I have fundamentally misunderstood some properties or something. I have two examples that I would like to have clarified. The first one is:

"Compute the Fourier transform of $f(x) = \cos \left( \frac{\pi}{2} x \right) \chi_{[-1,1]}(x)$ and justify the equation $\widehat f = \widehat{\cos\left( \frac{\pi}{2} x \right)} * \widehat{\chi_{[-1,1]}(x)}$."

My thoughts on this problem goes along the way of showing that the functions are contained in function spaces which we know satisfies the property that pointwise products and convolutions are interchangeable under the Fourier transform. Specifically the book lists the options: Both functions in $L^2$; One Schwartz function and a tempered distribution. Furthermore it is noted that $\widehat{S * T} = \widehat S \cdot \widehat T$ where both are tempered distributions and at least one of them has compact support. However I cannot seem to make sense of any of these properties and at this point I start to think that the "justify"-bit in the question just refers to showing that equality holds? The problem is then that I cannot manage to do that either. That might have something to do with me staring at this problem for too long.

The other problem is as follows:

"Compute the Fourier transform of $\text{pv} \left( \frac{1}{x} \right)$ from the relation $x \cdot \text{pv} \left( \frac{1}{x} \right) = 1$."

Also here I have the same problems as over trying to characterize the functions into a function space which allows me to do the operation that is glaring at me.

As always I would appreciate some hints in the right direction rather than have the solution printed out in front of me. The problems are found in Gasquet and Witomski "Fourier analysis and applications", chapter/lesson 33.

Thanks.

KryptoJon
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  • Do you know how to prove that any $L^p$ function defines a tempered distribution? This will help justify the first one. As for the second, the function $x$ is also a tempered distribution since it is continuous and of polynomial growth (in fact, it's a polynomial). Here are some hints to prove these statements: http://www2.math.uni-paderborn.de/fileadmin/Mathematik/People/ckaiser/Fourieranalysis/Vorlesungsinhalte/FourierAnalysis_2.pdf – dls May 16 '14 at 18:51
  • For the first statement I would use that $L^p$ sits inside of $L^1_{loc}$ and therefore can be represented as a distribution. To show the convergence property I would use Hölder's inequality. $x$ is a tempered distribution since it is slowly increasing and by properties of the Schwartz space. In the first case we then have two tempered distribution where one has compact support. But that is the "wrong" equality with regards to what I wrote in the first post? – KryptoJon May 16 '14 at 20:17
  • any progress on this question? – Conrado Costa Nov 03 '16 at 08:51

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