The basis functions for linear B-splines are: \begin{equation*} \begin{aligned} & B_0(u) = (1-u)\\ & B_1(u) = u. \end{aligned} \end{equation*}
For quadratic B-plines: \begin{equation*} \begin{aligned} & B_0(u) = u^2/2\\ & B_1(u) = (-2u^2+2u+1)/2\\ & B_2(u) = (u^2-2u+1)/2. \end{aligned} \end{equation*}
For cubic B-splines: \begin{equation*} \begin{aligned} & B_0(u) = (1-u)^3/6\\ & B_1(u) = (3u^3 - 6u^2+4)/6\\ & B_2(u) = (-3u^3 +3u^2+3u+1)/6\\ & B_3(u) = u^3/6. \end{aligned} \end{equation*}
Can anyone please explain how are these forms derived for the basis functions given the order of the B-splines?
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