I am working on a certain problem: say we have a function $f$ defined on an interval $[a,b]$ and we want the spline $Sf$ of order $k$ (and knots needed for the B-splines are in $[a,b]$) to agree to $f$ and its first $k-1$ derivatives at $a$. In doing so, I came across the following:
Suppose we have a system of the form ($j \in \mathbb{Z}$ fixed) \begin{align*} f^{(m)}(a) = \sum_{i = {(j-1)k+1}}^{jk} \alpha_i^{(m+1)} B_{i,k-m}(a), \quad m = 0, 1, \ldots, k-1, \end{align*} where $B_{i,r}$ is a B-spline of order $r$ and $\alpha_i^{(1)} = \alpha_i$ and $\alpha_i^{(m+1)} = (k-m)\cdot \dfrac{\alpha_i^{(m)} - \alpha_{i-1}^{(m)}}{t_{i+k-m} - t_i}$ for $m >0$.
Is there a neat way to formulate this into a matrix equation with the $\alpha_i$ as the unknowns?