Is it common to impose the sparsity on the Fourier coefficient itself? In compressive sensing, I see many works to impose the sparsity on the wavelet coefficients (e.g., by minimizing the L1 norm of such coefficients.) Another example in MRI is to impose sparsity on the gradient of the unknown image (total variation).
However, I'm interested in imposing the sparsity on the Fourier coefficients itself. I couldn't find for the moment. If there is any reference, could anyone let me know? Or, I'm not sure whether it doesn't make sense.
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).