= B-spline
{c}
{title2=$N_{i,k}$}
{wiki}
An order-$k$ <B-spline> $N_{i,k}$ is a local, nonnegative <spline> determined by $k+1$ consecutive entries $t_i,\ldots,t_{i+k}$ of a nondecreasing <spline knot sequence>. Its <polynomial> pieces have degree at most $k-1$, and its <support> is contained in $[t_i,t_{i+k}]$. The <Cox-de Boor recurrence> gives a constructive definition from interval indicators.<Math CS 887: Approximation Theory by Carl de Boor>[definition and recurrence (84)–(86), printed pp.53–54]{ref}
For example, the order-two <B-spline> with <spline knots> $0,1,2$ is the triangular hat
$$
N_{0,2}(x)=\begin{cases}x,&0\leq x\leq1,\\2-x,&1\leq x\leq2,\\0,&\text{otherwise}.\end{cases}
$$
Only nearby <B-splines> are nonzero at a given point. This local <support> makes <spline interpolation> <matrices> and <spline> <Gram matrices> sparse. On the basic knot interval, a completed <B-spline> <basis> has <partition of unity>; <spline knot> multiplicities determine the smoothness across joins.
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