Is it common to impose the sparsity on the Fourier coefficient itself?
This question is typically the subject of a paper like Robust Uncertainty Principles: Exact Signal Reconstruction From Highly Incomplete Frequency Information, Emmanuel J. Candès, Justin Romberg, 2006:

This paper considers the model problem of reconstructing an object
from incomplete frequency samples. Consider a discrete-time signal
$f\in \mathbb{C}^N$ and a randomly chosen set of frequencies $\Omega$.
Is it possible to reconstruct from the partial knowledge of its
Fourier coefficients on the set $\Omega$?

From citations within, and the quantity of citing papers, there is a possibility to find many useful references. Typically, a potential issue with non-local Fourier décompositions is that discontinuities in signals/images induce slow decay in frequency coefficients. Wavelets are usually better at dealing with sharp variations.
However, some modalities of imaging or spectroscopy are indeed acquired in a Fourier domain, and in practice, the compressibility of data (decaying coefficients) can suffice (over strict sparsity).