Does anyone know what the Fourier transform (in the sense of distributions) of $$ f(x) = (x^2  1)^{1/2}x, \quad x\ge 1, $$ and $f(x) = 0$ otherwise, is?
First of all observe that
$$ f(x)=\frac{1}{3}\frac{d}{dx} (x^21)^{\frac{3}{2}}_+, $$
where for any real number $t$ we set $t_+=\max(t,0)$. Thus it suffices to compute the Fourier transform of $(x^21)^{\frac{3}{2}}_+$.
In Section 2.5 Chapter 2 of the book by Gelfand and Shilov, Generalized Functions, vol.1, Academic Press 1964, the authors compute the Fourier transform of $(ax^2+bx+c)^\lambda_+$. Your example corresponds to Case (3) discussed there. More precisely the Fourier transform of $(x^21)^\lambda_+$ is the function
$$\Gamma(\lambda+1)\sqrt{\pi}\left\frac{\xi}{2}\right^{\lambda\frac{1}{2}}\frac{\cos\pi(\lambda+\frac{1}{2}) J_{\lambda\frac{1}{2}}(\xi)J_{\lambda+\frac{1}{2}}(\xi)}{\sin \pi(\lambda+\frac{1}{2})}, $$
where $J_\alpha$ denotes the Bessel function of order $\alpha$.

3$\begingroup$ What happens with the $\textrm{sin}\pi (\lambda+1/2)$ for $\lambda=3/2$? $\endgroup$ – Robert Haslhofer Apr 4 '12 at 0:22

2$\begingroup$ I thought nobody was going to ask this embarrassing questions. Here are two possible answers. You either let $\lambda\to\frac{3}{2}$ in the above formula or, better yet, consult the above reference where there is an alternate description involving the Bessel functions $N_\alpha$.. I did not include that formula since I would have had to explain what $N_\alpha$ is. $\endgroup$ – Liviu Nicolaescu Apr 4 '12 at 9:07

1$\begingroup$ Liviu, thank you so much for your response. I believe $N_\alpha $ is the same as $Y_\alpha $, i.e. the Bessel function of the second kind, which is what I got by taking the limit. And thanks Robert for bringing up the discussion of the limiting cases. $\endgroup$ – flavio Apr 4 '12 at 10:01

1$\begingroup$ You are absolutely right. The label $N_\alpha$ is an old fashion one that survived mainly behind the Iron Curtain and it reflects the fact that $Y_\alpha=N_\alpha$ is sometime the Neumann function. $\endgroup$ – Liviu Nicolaescu Apr 4 '12 at 11:51