I have four questions about splines. Any help would be greatly appreciated.
1) Boundary conditions for cubic spline interpolation to a set of data
$a=x{}_{1}<x2<…<x_{m} , $
like for example $not-a-knot$ or $Natural$ give respectively the knot sequences:
$t_{NK}=(x_{1},x_{1},x_{1},x_{1},x_{3},x_{4},…x_{m-2},x_{m},x_{m},x_{m},x_{m})$ and
$t_{N}=(x_{1},x_{1},x_{1},x_{1},x_{2},x_{3},…x_{m-1},x_{m},x_{m},x_{m},x_{m})$.
But if I want to interpolate with periodic end conditions:
$D^{j}g(a)=D^{j}g(b),$ for $j=1,2$ , then what is the corresponding knot sequence (and why) ?
2) Suppose I want to interpolate a function defined on a grid of size $h$.
What does it mean, mathematically, that cardinal B-splines have $h-independent$ $ L^{\infty} $$norm$ (and why)?
3) What is exactly a "cardinal" spline ? Are splines with the above knot sequences considered as cardinal splines ?
4) Would there be somewhere I could find an algorithm that can construct a parametric tensor product cubic cardinal B-spline interpolant (if I am not saying something wrong), that is fast for evaluation of the constructed interpolant (and ideally the matlab code) ?
Best Answer
(1) $(x_1, \ldots, x_m)$, to get the right dimension for the spline space (to match the number of constraints).
(2) It means that the cardinal b-splines have $L_\infty$ norm (maximum absolute value) that is independent of the grid size, $h$. I'm not sure why this would be the case. Need more context to explain further.
(3) A cardinal spline is one that has the value 1 at one interpolation point, and zero at all others. So, it's a statement about the values of the spline function, not about its knots. So, it doesn't make sense to ask whether or not the splines you mentioned are cardinal splines, since you didn't say anything about values.
(4) Some time ago there was a "spline toolkit" in Matlab. The code was written by Carl deBoor, based on his book "A Practical Guide to Splines" (which I highly recommend). Looking at the Matlab web site, the spline toolkit seems to have disappeared, and they are now talking about a "Curve Fitting" package, instead. But, anyway, it looks like a lot of the original code/functions are still present. For tensor product interpolation, look at this document, or look at deBoor's book.