Splines and Finite Element Spaces

Splines are piecewise polynomial functions that have certain “regularity” properties. These can be defined on all finite intervals, and intervals of the form (−∞, a], [b,∞) or (−∞,∞). We have already encountered linear splines, which are simply continuous, piecewise-linear functions. More general splines are defined similarly to the linear ones. They are labeled by three things: (1) a knot sequence, ∆; (2) the degree k of the polynomial; and, (3) the space Cr, the level of differentiability of the whole spline. The knot sequence is where the polynomial may change. For a linear spline defined on [0, 1], the knot sequence ∆ = {x0 = 0 < x1 < x2 < · · · < xn = 1} is where one linear polynomial meets another. Since the polynomials are linear, k = 1. Finally, since the he linear splines are continuous, they are in C0[0, 1], so r = 0.