Linear precision at Greville abscissae (${\sum }_i{\xi }_i{N}_{i,d}(t)=t$)

For an open clamped spline knot sequence and degree $d\ge 1$, the Greville abscissae ${\xi }_i=d^{-1}{\sum }_{r=1}^d{t}_{i+r}$ reproduce the parameter. The B-spline differentiation formula gives derivative coefficients $d({\xi }_i-{\xi }_{i-1})/(t_{i+d}-t_i)=1$. Their lower-degree partition of unity makes the derivative of the weighted sum equal to one. The left endpoint value is the left parameter endpoint, fixing the integration constant to zero. This proves linear precision even with unequal knot intervals.