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 $\Vert f-Pf{\Vert }_{\infty }\le (1+{\Lambda }_{\mathbf{x}}){\inf }_{s\in \mathrm{range}P}\Vert f-s{\Vert }_{\infty }$.