# Sensitivity of splines to control points

Let $$X\in\mathbb{R}^{n\times d}$$ 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 $$f$$?

• You probably know that the value of a cubic spline, at some fixed evaluation points $x_1,\ldots,x_m$, is a linear transformation of the values at the control points, with the transformation defined by the "hat matrix" of the spline. So at least some of your sensitivity questions are perhaps answered by inspecting the hat matrix. – Jukka Kohonen Jun 10 at 20:52