L1 / L infinity Projections (minimization)

Hi, I'm trying to project a function onto a subspace (specifically, sin(pi*x) onto L1 or Linfinity with a convexity constraint) and I'm having trouble accomplishing this. From what I've read it seems like I should use linprog but I can't quite see how to formulate this minimization in the given norm into what linprog wants as input.

What I've tried so far, for L1 at least, is to add additionally inequality constraints in the form eye(N,N)*x <= sin(pi*x) + epsilon (chosen beforehand) but this feels a bit wonky and the results don't agree with that in the paper I am trying to reproduce. Anyone have a suggestion?

Thanks!