= Schoenberg–Whitney theorem
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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 $k\geq2$, is invertible exactly when
$$
t_i<x_i<t_{i+k}\qquad(1\leq i\leq n).
$$
These 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.<Math CS 887: Approximation Theory by Carl de Boor>[Schoenberg–Whitney theorem (131), printed p.67]{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.
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