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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?

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up vote 16 down vote accepted

First of all observe that

$$ f(x)=\frac{1}{3}\frac{d}{dx} (x^2-1)^{\frac{3}{2}}_+, $$

where for any real number $t$ we set $t_+=\max(t,0)$. Thus it suffices to compute the Fourier transform of $(x^2-1)^{\frac{3}{2}}_+$.

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 $(x^2-1)^\lambda_+$ 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 $\alpha$.

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What happens with the $\textrm{sin}\pi (\lambda+1/2)$ for $\lambda=3/2$? – Robert Haslhofer Apr 4 '12 at 0:22
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. – Liviu Nicolaescu Apr 4 '12 at 9:07
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. – flavio Apr 4 '12 at 10:01
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. – Liviu Nicolaescu Apr 4 '12 at 11:51

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