calculating b splines with non-distinct knots

Question

How can one calculate B splines with non distinct knots. I know about carl de boor algorithm, but that is to my knowledge only applied to non distinct knots. We can assume the knots are in increasing order. In particular im looking to find $B_0^4(x)$ with knots being $x_0 = 0 \ x_1 = 0 \ x_2 = 1 \ x_3 = 2 \ x_4 = 2$ But how do you do it generally?

Answer 1

Just use deBoor's algorithm. It doesn't require distinct knots.

However ...
I assume that $B_0^4(x)$ means the b-spline basis function of order four (degree three) that is non-zero over the interval between knots $t_0$ and $t_4$. If my assumption is correct, you're in trouble. A basis function of order $4$ depends on six knots, and you have only five knots.

Answer 2

It will probably not solve your question, but perhaps you could do something like this?

Partition the family of points (size $n$) into families with only equal elements. Let $min$ be the minimal distance between any two of these families. For every such family (of equal points):
Choose for every point a close "representative point" (with a distance $ < min/2n$). This can be done in a manner such that every member of the current gets his very own, unique representative point.
Now, halve the distances between each point and its representative. Iterate. Calculate the limit.

It is not very well elaborated and certainly not easy to compute manually, but I'm interested wether in principle, this idea would work.