  In this paper, we investigate the global properties of Fourier multipliers in
the setting of nonharmonic analysis of boundary value problems. We give
necessary and sufficient conditions for a Fourier multiplier to be globally
hypoelliptic and also to be globally solvable. As an application, we consider
operators on $[0,1]^2$ with non-periodic boundary conditions and we obtain
results that extend what is already known in the periodic case.
