Fourier transforms and composite functions

Question

Suppose I have some bounded, smooth function $\psi(\vec x)$ such that its Fourier transform $\tilde\psi(\vec k)=0 $ for $|\vec k|\leq m $, for some bound $m\in\mathbb R^{>0}$. What are the necessary and sufficient conditions to ensure that for any smooth function $f$, we have that the Fourier transform of $f(\psi(x))$ also vanishes for $|\vec k|\leq m$.