= Optimal spline coefficient interpolation sites
{title2=$\min_{\mathbf x}\|A_{\mathbf x}^{-1}\|_\infty=\kappa(\mathcal S)$}
The equality holds when an alternating unit-norm <spline> $s_*$ has as many ordered extremal sites as the dimension, and those sites satisfy the <Schoenberg–Whitney theorem>. For every admissible set, sampling has <operator norm> at most one, giving $\kappa(\mathcal S)\le\|A_{\mathbf x}^{-1}\|_\infty$. At the alternating sites the <checkerboard inverse of a totally nonnegative matrix> identifies each inverse absolute row sum with the magnitude of the corresponding <coefficient> of $s_*$. The <Chebyshev spline coefficient and dual norm equality> therefore gives equality. This minimizes amplification into <B-spline> <coefficients>; it does not by itself prove optimality for the distinct <Lebesgue constant of interpolation>.
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