= Maximum-norm bound for spline projection
{title2=$\|P\|_\infty\leq\|G^{-1}\|_{\ell^\infty}$}
For the orthogonal <B-spline> projector $P$, the normal equations in the <mixed-normalization spline Gram matrix> give $Ga=r$, with $r_i=(M_i,f)$. Positivity and unit integral of $M_i$ imply $\|r\|_{\ell^\infty}\leq\|f\|_\infty$. Nonnegativity and subpartition of unity of $N_i$ give $\|\sum_ia_iN_i\|_\infty\leq\|a\|_{\ell^\infty}$. Hence $\|P\|_\infty\leq\|G^{-1}\|_{\ell^\infty}$, where the matrix norm is maximum absolute row sum. The range is the spline space, which lies in $C[0,1]$ only for continuous splines.
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