= Linear-spline mixed Gram matrix
{title2=$g_{ii}=2/3,\quad\sum_{j\ne i}|g_{ij}|\le1/3$}
For the distinct-knot linear <B-spline> hats, let $h_i=t_{i+1}-t_i>0$. The <mixed-normalization spline Gram matrix> has entries
$$
g_{ii}=\frac23,\quad g_{i,i-1}=\frac{h_i}{3(h_i+h_{i+1})},\quad g_{i,i+1}=\frac{h_{i+1}}{3(h_i+h_{i+1})},\quad g_{ij}=0\quad(|i-j|\ge2).
$$
Only indices in the finite <basis> are retained. This <matrix> has a row <strict diagonal dominance> margin at least $1/3$, so the <inverse infinity-norm bound from diagonal dominance> gives $\|G^{-1}\|_{\ell^\infty}\le3$. Equidistant <spline knots> give off-diagonal entries $1/6$. Both pieces of every hat are retained; repeated endpoint <spline knots> require different boundary formulas.
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