Schoenberg–Whitney theorem

For strictly increasing spline knots $t_1<\cdots <t_{n+k}$ and interpolation sites $x_1<\cdots <x_n$, the order-$k$ B-spline collocation matrix $A_{ij}=N_{j,k}(x_i)$, with $k\ge 2$, is invertible exactly whenThese conditions place each site in the open support of its corresponding B-spline, and are equivalent to $N_{i,k}(x_i)>0$ in this simple-knot setting. They characterize existence and uniqueness of spline interpolation for all data.^[ref]

With repeated clamped endpoint spline knots, endpoint conventions matter: strict inequalities at the outer endpoints are not necessary. For example, the order-two clamped sequence $(0,0,1,1)$ gives the basis $1-x,x$ on $[0,1]$. Sampling at $0,1$ gives the identity matrix, even though the corresponding outer strict inequalities fail. One must use the version allowing nonzero endpoint B-spline values instead of importing the simple-knot inequalities unchanged.