Multirate periodic systems and some related constrained analytic function interpolation problems are studied in this paper. After showing how to convert a general multirate periodic system to an equivalent linear time invariant (LTI) system with a structural constraint, we formulate some analytic function interpolation problems with such a constraint that can find various applications in the study of multirate and periodic systems. Both the solvability conditions and characterization of all solutions are presented to these constrained interpolation problems.