# Relation of Cox-deBoor recurssion and Convolution (b-spline basis)

Consider the Cox-deBoor recursion formula for producing b-spline basis functions given a knot vector:

$N_{i,0}(u)=1$ if $u_i\leq u < u_{i+1}$ otherwise, $=0$

$N_{i,p}(u)=\frac{u-u_{i}}{u_{i+p}-u_{i}}N_{i,p-1}(u)+ \frac{u_{i+p+1}-u}{u_{i+p+1}-u_{i+1}}N_{i+1,p-1}(u)$

Now, I read that b-Splines can also be produced using recursive convolution instead of the recursion formula above. For example, see this page: http://www.chebfun.org/examples/approx/BSplineConv.html

Can someone explain if these two are related in any way, I am just not seeing it? And if so, how can I apply convolution to the knot spans of a knot vector to produce the same b-spline basis that we'd get by using the Cox-deBoor formula?

• I'd guess that the convolution formula works only in the case of equally-spaced knots. Just a guess, though. And, for this case, the algebra simplifies drastically, and I would think you could just prove by brute force that both processes produce the same functions. – bubba Sep 17 '16 at 2:09
• Does this help: math.stackexchange.com/questions/114813/… – bubba Sep 17 '16 at 2:13