  In this note, we develop Fourier approximation methods for the solutions of
first-order nonlocal mean-field games (MFG) systems. Using Fourier expansion
techniques, we approximate a given MFG system by a simpler one that is
equivalent to a convex optimization problem over a finite-dimensional subspace
of continuous curves. Furthermore, we perform a time-discretization for this
optimization problem and arrive at a finite-dimensional saddle point problem.
Finally, we solve this saddle-point problem by a variant of a primal dual
hybrid gradient method.
