On bases of cardinal functions and their role in approximate sampling methods

This article discusses the structure of various shift-invariant spaces of cardinal functions generated by a single kernel, and their role in some approximate sampling methods. Particularly, conditions on the generating kernel φ are given which imply that the associated L
_p
shift-invariant space coincides with the shift-invariant space for the cardinal function L
_φ
generated by the kernel. Additionally, we discuss how such spaces are used in summability methods for the classical sampling series.