I would like to sparsely represent a linear function/ramp in the Fourier domain. In an attempt to improve the sparsity, I have zero padded it. With this padding, it is possible in the example I tried to represent the function with just one non-zero component in the Fourier domain. When I try to solve it as an inverse problem, however (using the L1 norm to make it tractable), it finds a different, substantially less sparse solution. Is there any way to improve this? The function will not always be straight, and may sometimes need more than one component to represent it.
My current approach is to use Basis Pursuit to minimise $|m|_1$ such that $P^HF^Hm = d$, where $m$ is the model in the Fourier domain (that I wish to be sparse), $P^H$ is the adjoint of the padding operator (so it discards the padded area), $F^H$ is the adjoint/inverse Fourier transform, and $d$ is the linear function/ramp to match.
The image shows the True signal ($d$), the Desired signal that is sparse in the Fourier domain and that matches the data, and the Actual (not very sparse in the Fourier domain) signal that I get when I use Basis Pursuit. The Actual signal does have a lower L1 norm than the Desired signal, so Basis Pursuit did its job, but it is not the solution that I want.
