# B-Spline Question

Show that the cubic B-Spline with integer knots can be written as

$$s(x) = \frac{1}{6}\left [ x^3 \; x^2 \; x \; 1\right ]\begin{bmatrix} -1 &3 & -3 & 1 \\ 12 &-29 & 12 & 0 \\ -48 & 60 & -12 & 0\\ 64 & -44 & 4 & 0 \end{bmatrix}\begin{bmatrix}c_3\\ c_2 \\ c_1 \\ c_0\end{bmatrix}$$

$= b_{30}c_0 + b_{31}c_1 + b_{32}c_2+ b_{33}c_3$

where

$$B_{0}^3 = \begin{bmatrix}b_{30} \; (0 \le x < 1) \\ b_{31}\; (1 \le < 2) \\ b_{32} \; (2 \le x < 2) \\ b_{33} \; ( 3 \le x < 4) \\ 0 \; (\text{otherwise})\end{bmatrix}$$

I'm not sure how to start this problem off and solve it. Usually, I try and show that $s(x)$ satisfies the conditions to be a B-Spline. However, I don't know how to do that here.