Schoenberg spline operator ($Sf={\sum }_{i}f({\xi }_i)N_{i,k}$)

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.

For nonnegative partition-normalized order-$k$ B-splines, let $Vf(t)={\sum }_{i}f({\tau }_i)N_i(t)$ with each sample point inside $[t_i,t_{i+k}]$. If $N_i(t)$ is nonzero, both $t$ and ${\tau }_i$ lie in an interval of length at most $k|\Delta |$. Their function values differ by at most the modulus of continuity at that length. Summing with nonnegative weights of total one proves the displayed bound. With fixed order and vanishing maximum knot gap, uniform continuity gives uniform convergence. The same argument works for bounded piecewise-continuous outputs if full interior knot multiplicities are allowed; their individual continuity should not then be asserted.