 Schoenberg spline operator

For an order-$k$ partition-normalized B-spline basis with $k\ge 2$, the standard Schoenberg spline operator samples at the Greville abscissae:It is a positive linear operator on continuous functions on the basic knot interval. Partition of unity gives $S1=1$, and linear precision at Greville abscissae gives $Sx=x$. These reproduction properties rely on the specified samples; choosing arbitrary points in the individual supports does not generally reproduce a linear function.^[ref]

Local support gives a direct approximation bound. If every active B-spline at $x$ has $|{\xi }_i-x|\le h$, then positivity and partition of unity give $|Sf(x)-f(x)|\le \omega (f,h)$, where $\omega$ is the modulus of continuity. Hence mesh refinement with shrinking active support widths yields uniform convergence for every continuous function, while avoiding a global interpolation solve.