Torwards a general formulation for over-sampling and under-sampling

We investigate over-sampling and under-sampling scenarios under the formulation of a generalized sampling model. Usually, these scenarios are described in the context of the so-called Shannon´s sampling theorem. This can be easily extended to more general settings. We first revisit a conventional definition of over-sampling and under-sampling in a general setting, and point out that the definition consists of two conditions. To treat them separately, we introduce the two notions of `perfect reconstruction´ and `redundant sampling.´ We show that these concepts are geometrically characterized by using sampling and reconstruction spaces. Then, we show that there appear four types of scenarios, which includes the conventional over-sampling and normal sampling, and further two types of under-sampling scenarios. The second type is more counter intuitive because it satisfies both non-perfect reconstruction and redundant sampling scenarios. We illustrate this last scenario by a practical example that involves cyclic B-spline functions.