||Periodic and multirate systems have wide applications in control, signal processing, communication, econometrics and numerical mathematics. The reason for studying periodic and multirate systems may be due to their power in modelling physical systems with inherent features like periodic behavior changes, seasonal operating environment, nonuniform information exchange pattern, multirate sampling, etc., or due to the fact that they can often achieve objectives that cannot be achieved by single rate linear time invariant (LTI) systems. However, the theory of periodic and multirate systems in the literature is mostly problem-specific and the tools developed are not as complete and comprehensive as those for single rate systems. In this thesis, we will present a unified analysis and synthesis approach for periodic and multirate systems. First we give the setup of multirate periodic (MP) systems which covers many familiar classes of systems including periodic systems, dual rate systems and sampled-data systems as special cases. Using the technique of lifting, we show that an MP system can be converted to an equivalent LTI system with a block lower triangular feedthrough term. On the other hand, analytic function interpolation is a very powerful tool in a variety of engineering problems such as in control, circuit theory, digital filter design and spectral estimation for LTI systems. Motivated by this reason and the fact that an MP system can be converted to an equivalent LTI system satisfying a structural constraint, we propose some constrained analytic function interpolation problems pertinent to MP systems, which play the same role as their unconstrained counterparts to LTI systems. Some necessary and sufficient solvability conditions and the parameterization of all solutions are presented. Using the results on constrained analytic function interpolation problems, we further study the robust model validation problem for MP systems. Both frequency domain and time domain validation tests are carried out for MP uncertain systems. After giving the definition of the v-gap metric of two MP systems, we study the robust stabilization of MP systems with v-gap metric uncertainty. The optimal robust stability margin and an observer-based suboptimal controller are presented explicitly. In summary, this thesis formulates and solves some constrained analytic function interpolation problems and uses them to study robust robust model validation and stabilization for MP systems.