B-Spline

Definition

B-Splines are basis functions for spline function in which order is the same, meaning that all possible spline function can be expressed as a linear combination of B-splines.

Consider the knots $t_0,t_1,\dots t_m$ sorted into non-decreasing order. For the given sequence of knots, the B-splines of degree $p=0$ is defined as $B_{j,0}(x)=\{\begin{array}{ll}1& t_j\le x\le t_{j+1}\\ 0& \text{otherwise}\end{array}$ And the higher $(p+1)$-degree B-splines are defined by recursion $B_{j,p}(x)=\frac{x-t_j}{t_{j+p}-t_j}{B}_{j,p-1}(x)+\frac{t_{j+p+1}-x}{t_{j+p+1}-t_{j+1}}{B}_{j+1,p-1}(x)$

The order $n$ spline function on a given set of knots can be expressed as $s(x)=\sum _{j=1}^m{\gamma }_j{B}_{j,p}(x)$