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!
Best 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.