With the same number of knots, will the cubic Truncated power basis (cubic spline) produce the same results as B-spline?

Question

I wrote a thesis on expanding a model with cubic truncated power basis and B-spline. In the defense, one professor pointed out that I should get the same results with the two methods when the number of knots are the same. She said that she referred to de Boor (2001). Unfortunately, the truncated power basis part in de Boor (2001) does not makes sense to me. I referred to Fahrmeir et al. (2013) and Hastie et al. (2009), but there is no such conclusion.

Could anybody please clarify it? Thank you so much.

Answer 1

I do not know if you are still interested in this but I might have an idea about the remark you received. Indeed B-splines can be computed as (divided) differences of truncated power bases (as you pointed out, the reference for this is de Boor, 2001). Formulas in this context become easily ugly (especially with free-knots) but if we limit our analysis to equally-spaced knots, it is possible to demonstrate that $$ B^{p}_{j}(x) = (-1)^{p+1} \frac{\Delta^{p+1} f_{j}^{p}(x)}{h^{p} p!} $$ where $B^{p}_{j}(x)$ is a $p$-order B-spline basis, $\Delta^{p+1}$ is a $(p+1)$-order difference operator, $f_{j}^{p}(x)$ is a truncated power function of order $p$ and $h$ is a constant equal to the distance between two consecutive knots.

You will find this discussion interesting: Splines - basis functions - clarification. At the link you will also find a small example with a computer code. I hope my answer helps (at least a bit).