Let $f\in C([a,b])$, and let $(\tau_m)$ be a sequence of knot partitions of $[a,b]$ whose mesh size tends to $0$. For each fixed degree $r\ge 0$, there exist splines $s_m$ of degree at most $r$ subordinate to $\tau_m$ such that
where $h_m$ is the mesh size of $\tau_m$, $\omega(g,h)=\sup\{|g(x)-g(y)|:|x-y|\le h\}$ is the modulus of continuity, and $C_{r,k}$ depends only on $r$ and $k$. If $f\in C^{r+1}([a,b])$, then a corresponding endpoint estimate has the form