The stability of space plasmas in which the macroscopic variables vary in the direction l perpendicular to the magnetic field is usually studied by means of the local approximation, in which the fluctuating potential is assumed to be independent of x. To remove this approximation, we study a nonlocal problem in which the background density, and hence the fluctuating potentials, are periodic in x. From the linear Vlasov/Poisson equations we derive a set of coupled, linear, homogeneous, algebraic equations relating the Fourier amplitudes of the eigensolution. We solve the equations numerically and find a hierarchy of exact eigenmodes characterized by different growth rates and spatial structures. At points where the density gradients is locally zero, the mode amplitude is generally several orders of magnitude lower than the peak amplitude. We study the dependence of these solutions on the parameters which define the background density perturbation and demonstrate a correspondence between these nonlocal results and those obtained with the local approximation. This allows us to illuminate the conditions under which the local theory is a valid approximation and permits us to extend its range of useful application.