= Solution
Fix $i$. If $t<t_i$, all knot values of the function $x\mapsto(x-t)_+^{k-1}$ agree with a <polynomial> of degree $k-1$. Its order-$k$ <divided difference> vanishes. If $t>t_{i+k}$, all the knot values are zero, so the same <divided difference> again vanishes. Thus \b[the support is finite]:
$$
\boxed{\operatorname{supp}M_i\subseteq[t_i,t_{i+k}],\qquad\operatorname{supp}N_i\subseteq[t_i,t_{i+k}].}
$$
The <support and Gram bandwidth of B-splines> also holds at repeated knots: the same conclusion follows with the standard limiting convention. If $j\geq i+k$, then $t_j\geq t_{i+k}$, so the two supports have no positive-length overlap. An endpoint intersection contributes zero to the integral defining $G_{ij}$. The case $i\geq j+k$ is identical. Hence
$$
\boxed{G_{ij}=0\quad\text{when }|i-j|\geq k,\qquad d=k.}
$$
For strictly increasing knots, <B-splines> whose indices differ by $k-1$ overlap on a positive-length interval and their product is positive there, so this universal bound is sharp. In standard matrix terminology the half-bandwidth is $k-1$ and the full bandwidth is $2k-1$; the paper's integer $d$ is $k$.
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