Support and Gram bandwidth of B-splines
= Support and Gram bandwidth of B-splines
{title2=$G_{ij}=0\quad(|i-j|\geq k)$}
An order-$k$ <B-spline> is supported on $[t_i,t_{i+k}]$. Below this interval its truncated-power knot data agree with a degree-$k-1$ polynomial, annihilated by the order-$k$ <divided difference>; above it all data vanish. Therefore its ordinary or mixed <Gram matrix> has entries zero when $|i-j|\geq k$. The half-bandwidth is $k-1$, and for strictly increasing knots this bound is sharp.