= Chebyshev spline coefficient and dual norm equality
{c}
{title2=$|a_i^*|=\|\mu_i\|$}
Suppose $\|s_*\|_\infty=1$ and $s_*(x_j)=(-1)^j$ at increasing sites satisfying the <Schoenberg–Whitney theorem> conditions. For the <B-spline collocation matrix> $A$, the coefficient <linear functional> in the <dual basis> is $\mu_i(s)=\sum_j(A^{-1})_{ij}s(x_j)$. Therefore $\|\mu_i\|\leq\sum_j|(A^{-1})_{ij}|$. The <checkerboard inverse of a totally nonnegative matrix> makes $\mu_i(s_*)=(-1)^i\sum_j|(A^{-1})_{ij}|$, attaining that bound. Since $\mu_i(s_*)=a_i^*$,
$$
|a_i^*|=\|\mu_i\|=\sum_j|(A^{-1})_{ij}|.
$$
The conclusion is an extremal property of an alternating unit-norm <spline>, not a claim that every bounded <spline> has these extremal coefficients.
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