Cox–de Boor recursion

Remarks

The proof applies the Leibniz rule for divided differences to the product $(t - x)_+^{k-1} = (t - x) \cdot (t - x)_+^{k-2}$ and uses the recursive definition of divided differences. Many texts take the recursion itself as the definition of B-splines, and then the divided difference formula becomes a theorem. Example: with knots $0, 1, 2$, the recursion gives $B_{0,2}(x) = x$ on $[0, 1)$, $2 - x$ on $[1, 2)$ and $0$ elsewhere, the hat function. The recursion shows at once that $B_{i,k}$ is nonnegative, vanishes outside $[t_i, t_{i+k})$, and is a piecewise polynomial of degree at most $k - 1$. It is numerically stable, since for $x$ in the support both coefficients lie in $[0, 1]$.

Going deeper

For a student

The recursion, found independently by Cox and by de Boor in 1972, is the standard way to evaluate B-splines and their derivatives. Applying it to a whole spline gives de Boor's algorithm, which evaluates $\sum_j c_j B_{j,k}(x)$ by repeated convex combinations of the coefficients. From it one also derives the derivative formula for B-splines and the partition of unity $\sum_j B_{j,k}(x) = 1$ on the basic interval.