Fourier transform of f on the unit sphere

Question

My question is: do we have the right to compute the fourier transform of a function over the unit sphere? To be more precise, let $f\in L^1(\mathbb{R} ^n)$, $n\geq1$. Is the integral $$\int_{S^{n-1}}f(t)e^{-i\langle \omega|t\rangle} d\sigma(t), $$ make any sense? With of course $S^{n-1}$ denotes the unit sphere and $d\sigma$ is the measure over the sphere.

Answer 1

It does not make sense as is. An element of $L^1(\mathbb{R}^n)$ is not a function. It is an equivalence class of functions modulo equality away from (Lebesgue) measure zero sets. The sphere $S^{n-1}$ has measure zero, so it does not make sense to evaluate your $f$ on it. You need some regularity hypothesis on $f$, e.g., being continuous or being in a Sobolev space with a trace theorem to $S^{n-1}$.