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Hello,

May I ask how to define the fourier transform of a function that is not defined on the whole real line?

For example, what is the fourier transform of $\frac{1}{\sqrt{x}}$? And what is the fourier transform of $\frac{1}{\sqrt{10-x^2}}$?

I have to do these computations in a mathematical physics project, and these expressions come from physical systems like the harmonic oscillator. These real valued functions only make sense when $x$ is restricted to a certain interval of the real line. Can someone suggest me some good ways to handle this issue? Thanks!

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Why not just define the function to be zero outside the interval? Or, if the interval is bounded, why not use Fourier series instead? You need to specify in more detail why you want to use the Fourier transform; it changes certain functions into certain other functions, but without knowing what properties you seek, it is difficult to know exactly what you should be looking for. –  Zen Harper Jun 15 '11 at 4:52
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If the function $(10-x^2)^{-1/2}$ arises in some physical model, the chances are that this is implicitly understood to vanish outside the interval $[-\sqrt{10},\sqrt{10}]$, in line with what Zen suggests. On the other hand, sometimes one observes values on a finite interval but knows for some physical reason that it should be periodic (e.g. electric potential from a mains socket) and so in that case one should use Fourier series. –  Yemon Choi Jun 15 '11 at 5:52
    
Alternatively, you can analytically continue these functions to complex functions on the complex line minus some branch cuts. You should end up with a tempered distribution that depends on your choice of branch. –  S. Carnahan Jun 15 '11 at 11:00
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From the previous answers, you have seen that there are MANY different ways of defining this ill-defined Fourier transform. You say: "I have to do these computations". So you haveto make a choice. Please make sure that you know how to justify your choice. In other words, it's not enough justify the computation after the fact by saying "it provided the answer that I wanted to get". You need to understand why the particular choice that you made is the correct one to make. –  André Henriques Jun 15 '11 at 12:49
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1 Answer

Both examples are $L^1_{loc}$ functions which are bounded at infinity, thus are tempered distributions, that is continuous linear forms on the Schwartz space $\mathscr S$ of rapidly decreasing function. The definition of the Fourier transform on $\mathscr S'$ is $$ \langle \hat u,\phi\rangle_{\mathscr S',\mathscr S}= \langle u,\hat \phi\rangle_{\mathscr S',\mathscr S}. $$ From this definition, you get for instance that the Fourier transform of $x^{-1/2}1_{\mathbb R_+}(x)$ is an homogeneous distribution of degree $-1/2$. Let's do the explicit computation. For $z\in \mathbb C, \Re z>0$, $$ \int_0^{+\infty}x^{-1/2} e^{-2π z x} dx=(2π z)^{-1/2}\int_0^{+\infty}t^{-1/2} e^{-t} dt =(2 z)^{-1/2}=2^{-1/2}e^{-\frac{1}{2}Log z} $$ with $Log z=\int_{[1,z]}\frac{d\zeta}{\zeta}$ for $z\in \mathbb C\backslash\mathbb R_-$. We may as well extend that formula for $z=i\xi+0$ so that $$ Log z=\ln \vert \xi\vert+\frac{i\pi}{2}sign \xi $$ and the sought Fourier transform is $$ 2^{-1/2}\vert \xi\vert^{-1/2}e^{-\frac{i\pi}{4}sign \xi}. $$ That formal computation is well justified by the weak definition given at the beginning of the answer.

Bazin.

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