The natural cubic interpolating spline is the unique $C^2$, interpolating cubic spline, endowed with two extra boundary conditions. Obtaining this spline, denoted by $s(x)$, involves the inversion of a 5-diagonal matrix, and can therefore be shown to be nonlocal, i.e., every interpolation point $(x,f(x_i) )$ affects $s(x)$ at every point $x$. See here for example.
My questions: Is there any result indicating that the natural cubic spline is almost non-local, i.e., that as $|x_i - x|$ grows, the effect of $f(x_i)$ on $s(x)$ vanishes?
- The non-interpolating B-splines approximation can easily be local, but I am interested in the interpolating spline.
- Numerics indicate that, in practice, the natural cubic spline is almost local.