Assume that $$\hat f(x)= (2\pi)^{-n/2} \int_{\mathbb{R}^n} f(y) e^{-i\left<x,y\right>} dy$$ is the Fourier transform of a function $f$. What is $\hat f$ if $f(x)=|x|^{2-n}$?

  • $\begingroup$ $f(x)=|x|^{2-n}$ is not even integrable. $\endgroup$ – saz Nov 5 '14 at 19:16

Polar coordinates. $\int_{\mathbb R^n} |x|^{k} dx= \int_{S^{n-1}} \int_{0}^{\infty} r^{n-1}\cdot r^{k} \enspace dr d\theta$.

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  • $\begingroup$ The left-hand side equals $\infty$, right? $\endgroup$ – saz Nov 5 '14 at 19:18
  • $\begingroup$ @saz yes indeed $\endgroup$ – Haha Nov 7 '14 at 7:41

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