 Chebyshev spline coefficient and dual norm equality

Suppose $\Vert s_{\ast }{\Vert }_{\infty }=1$ and $s_{\ast }(x_j)=(-1{)}^j$ 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 $\Vert {\mu }_i\Vert \le {\sum }_j|(A^{-1}{)}_{ij}|$. The checkerboard inverse of a totally nonnegative matrix makes ${\mu }_i(s_{\ast })=(-1{)}^i{\sum }_j|(A^{-1}{)}_{ij}|$, attaining that bound. Since ${\mu }_i(s_{\ast })=a_i^{\ast }$,The conclusion is an extremal property of an alternating unit-norm spline, not a claim that every bounded spline has these extremal coefficients.