Calculating Fourier Transform of $1/|t|^n$

Question

I have found the Fourier Transform of $x(t)=|t|^{n}$ and i can't calculate the Fourier Transform of $x(t)=|t|^{-n}$. Any suggestions?

Answer 1

The Fourier transform is defined for tempered distributions. The function $x(t)=|t|^{-n}$ is not a distribution since it is not locally integrable. Therefore, it does not have a Fourier transform.

Loosely speaking: if the transform of $|t|^{-n}$ existed, it would be a constant function, because the Fourier transform of a radially symmetric homogeneous distribution of degree $d$ is a radially symmetric distribution of degree $-d-n$. With $d=-n$ this gives $0$. However, the constant function is the Fourier transform of the Dirac $\delta$. This leaves $|t|^{-n}$ out in the cold.

Related threads: ($n$-dimensional) Inverse Fourier transform of $\frac{1}{\| \mathbf{\omega} \|^{2\alpha}}$, What's the Fourier transform of these functions?.