A spline interpolates data—that is, fills the gaps between known points. Between two adjacent points a spline is an ordinary polynomial. The simplest spline is a connecting line (a linear spline), but here we consider quadratics.
Compressive sensing challenges us to think carefully about the sampling theorem. Remarkably, it is new (this century), accessible (without a Ph.D. in mathematics), and very useful in practice. For a first taste of compressive sampling see this blab (we’ve embedded some content below to whet your appetite.)