Sensitivity of splines to control points

Let $Xinmathbb{R}^{ntimes d}$ be a set of $n$ points in $mathbb{R}^d$, and let $f(X)$ be the operator that returns some spline interpolation of these points (say, cubic interpolation or Bezier curves). Is there some collection of results about the sensitivity of the interpolation w.r.t. the choice of control points? In other words, if $|cdot|_p$ is some metric on the control points and $|cdot|_c$ is some metric between curves, are there results about the operator norm of $f$?