Let $\phi:\mathbb{R}^n \to \mathbb{R}$ be a $C^{\infty}$ function with compact support.

What is the relationship between

$$ \widehat{\phi}(k) = \int e^{-2\pi i x \cdot k} \phi(x) dx, \quad k \in \mathbb{R}^n $$

and

$$ \widetilde{\phi}(\ell) = \int e^{-2\pi i |x| \ell} \phi(x) dx, \quad \ell \in \mathbb{R} $$

In particular, I would like to know what I can say about $\widehat{\phi}(k)$ if I know that $\limsup_{\ell \to \infty} |\widetilde{\phi}(\ell)| > 0$.

Edit:

In my attempt to simplify a problem, I have simplified it too much. As Willie Wong has pointed out, $\limsup_{\ell \to \infty} |\widetilde{\phi}(\ell)| > 0$ is never satisfied for $\phi$ smooth and compactly supported.

I'm actually interested in the situation where $\phi(x)dx$ is replaced by $d\mu(x)$, where it is indeed possible to have $\limsup_{\ell \to \infty} |\widetilde{\phi}(\ell)| > 0$.

theanswer. Given that $\tilde{\phi}$ is only sensitive to the spherical symmetric part of $\mu$, what you wrote pretty succinctly captures the difference. $\endgroup$