Irregular sampling and the theory of frames, I

Irregular sampling expansions are proved in an elementary way by an analysis of the inverse frame operator The expansions are of two dual types: in the first,the sampled values at irregularly spaced points are the coefficients; in the second, the sequence of sampling functions are irregularly spaced translates of a single sampling function. The results include regular sampling theory as well as the irregular sampling theory of Paley- Wiener, Levinson, Beutler,and Yao-Thomas.The use of frames also gives rise to a new interpretation of a aliasing.