Lebesgue constant of interpolation (${\Lambda }_{\mathbf{x}}={\max }_t{\sum }_i|{\ell }_i(t)|$)

If interpolation at distinct sites $x_i$ in a finite-dimensional space of continuous functions is unique, its cardinal functions satisfy ${\ell }_i(x_j)={\delta }_{ij}$. The interpolation linear map is $Pf={\sum }_{i}f(x_i){\ell }_i$ and has operator norm ${\Lambda }_{\mathbf{x}}$ in the supremum norm. The upper bound is the triangle inequality. For the reverse bound choose node values to be the signs of the cardinal functions at a point maximizing their absolute sum, and extend those values by a continuous function of norm one. The polynomial reproduction error bound gives $\Vert f-Pf{\Vert }_{\infty }\le (1+{\Lambda }_{\mathbf{x}}){\inf }_{s\in \mathrm{range}P}\Vert f-s{\Vert }_{\infty }$.

For a real $n$-dimensional vector space of continuous functions on a compact interval, choose sites maximizing the absolute evaluation determinant in any basis. Linear independence ensures some nonzero evaluation determinant, so a maximizer exists and has distinct sites. Replacing row $i$ by evaluation at $t$ multiplies the determinant by the cardinal function ${\ell }_i(t)$. Maximality implies $|{\ell }_i(t)|\le 1$, hence the Lebesgue constant of interpolation is at most $n$. This gives a useful interpolation set without asserting that determinant maximization minimizes the Lebesgue constant of interpolation.