Nodal derivative norm from Lagrange cardinal polynomials (${\sup }_{|p(t_i)|\le 1}|p^{(k)}(x)|={\sum }_i|{\ell }_i^{(k)}(x)|$)

At distinct interpolation nodes, differentiating Lagrange interpolation gives $p^{(k)}(x)={\sum }_{i}p(t_i){\ell }_i^{(k)}(x)$. The triangle inequality gives the displayed upper bound, attained by prescribing each nodal value to be the sign of the corresponding derivative weight. For increasingly ordered real nodes, the Markov interlacing lemma and monotonicity of polynomial critical points in their roots imply that $(-1{)}^i{\ell }_i^{(k)}(x)$ has at most one sign change, ignoring zero weights. Thus an extremizer has alternating nodal signs with at most one reversal of that alternating pattern. This reduces a choice among $2^{n+1}$ sign patterns to $n+1$ candidate polynomials.