Generalized Splines

In this chapter, we define bivariate splines. The term spline is often used synonymous with linear combinations of B-splines and hence piecewise polynomials. We will define splines in a less restrictive fashion to include, for example, trigonometric splines and functions generated by interpolating refinement algorithms. This will allow us to cover the shared underlying fundamentals once and for all. Specifically, splines are defined as continuous functions on a domain that is a topological space. This domain is the result of gluing together indexed copies of the unit square, and is locally homeomorphic to the domain of standard bivariate tensor product spline spaces — except at extraordinary knots where more or fewer than four unit squares join up. Consequently, we can focus on characterizing analytical and differential-geometric properties of such splines at and near these isolated singularities.