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?
Remarks:
- 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.
Let $S_i$ be the unique cubic spline such that $S_i(i) = 1$ and $S_i(j) = 0$ for $i \ne j$. These are sometimes called "cardinal splines". Obviously any cubic spline with breaks at the integers can be written as a linear combination of the $S_i$.
The "almost local" property follows from the fact that $S_i(x)$ decays exponentially as $|x-i|$ increases. There is a discussion here, but you can probably find more modern treatments, too.
Presumably, there is some connection with the fact that elements of the inverse of a banded matrix decay exponentially. Reference here.