= Local-support error bound for a Schoenberg spline operator
{title2=$\|Vf-f\|_\infty\le\omega(f,k|\Delta|)$}
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.
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