I need some help with the following question.

If f has a Fourier transform F(k), what is the Fourier transform of cos(x)f(2x+1).

I have made pretty much no progress on this. This seems straightforward if I were to be asked for the Fourier transform of cos(x)f(x), but I don't know in what way the "2x+1" will change the integral. Any help would be appreciated.

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    $\begingroup$ Would you know how to do it if it were $\cos((x-1)/2)) f(x)$? $\endgroup$ – Ian Apr 28 '16 at 3:36

The following can be found in a table of transforms: Let $f$ have the Fourier transform $\hat{f}$.

$x \mapsto f(ax)$ has Fourier transform $k \mapsto {1 \over |a|}\hat{f}({k \over a}) $.

$x \mapsto f(x-a)$ has Fourier transform $k \mapsto e^{-iak}\hat{f}(k) $.

$x \mapsto e^{i a x} f(x)$ has Fourier transform $k \mapsto \hat{f}(k-a)$.

The first two can be used to find the transform of $x \mapsto f(2x+1)$.

The last, along with the fact that $\cos x = {1 \over 2} (e^{ix} +e^{-ix})$, can be used to finish.


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