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.