Spatially periodic systems are of interest in problems of science and engineering and are typically described by partial (integro) differential equations with periodic coefficients. In this work we present tools for the analysis of such systems in the spatial-frequency domain.
   In Part I of this dissertation, we describe the basic theory of spatially periodic systems. We use the frequency lifting operation to represent a spatially periodic system as a family of infinitely-many coupled first-order ordinary differential equations. We describe the notions of stability and input-output norms for these systems, and give nonconservative results both for determining stability using the Nyquist criterion and for evaluating system norms.
   In Part II of this dissertation, we use perturbation methods to analyze the properties of spatially periodic systems. It is often physically meaningful to regard a spatially periodic system as a spatially invariant one perturbed by spatially periodic operators. We show that this approach leads to a significant reduction in the computational burden of verifying stability and estimating norms. Although perturbation methods are valid for small ranges of the perturbation parameter and may give conservative results, they can be very beneficial in revealing important trends in system behavior, identifying resonance conditions, and providing guidelines for the design of periodic structures.