= Schoenberg spline operator
{c}
{title2=$Sf=\sum_i f(\xi_i)N_{i,k}$}
For an order-$k$ partition-normalized <B-spline> <basis> with $k\geq2$, the standard <Schoenberg spline operator> samples at the <Greville abscissae>:
$$
Sf(x)=\sum_i f(\xi_i)N_{i,k}(x),\qquad\xi_i=\frac{t_{i+1}+\cdots+t_{i+k-1}}{k-1}.
$$
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>.<Math CS 887: Approximation Theory by Carl de Boor>[definition (124) and properties (125)–(127), printed p.65]{ref}
Local <support> gives a direct approximation bound. If every active <B-spline> at $x$ has $|\xi_i-x|\leq h$, then positivity and <partition of unity> give $|Sf(x)-f(x)|\leq\omega(f,h)$, 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.
Back to article page
