Simple-knot B-spline regularity ($N_{i,k}\in C^{k-2},\mathrm{supp}{N}_{i,k}=[t_i,t_{i+k}]$)

For distinct increasing knots and $k\ge 2$, the explicit divided difference formula writes an order-$k$ B-spline as a finite sum of truncated power functions of degree $k-1$. It is a piecewise polynomial function and is globally $C^{k-2}$. Below its first knot the order-$k$ divided difference annihilates a degree-$k-1$ polynomial; above its last knot all values vanish. Nonzero first and last pieces give the exact closed support. Order one gives interval indicators instead of continuous splines.