= Lebesgue constant of interpolation
{c}
{title2=$\Lambda_{\mathbf x}=\max_t\sum_i|\ell_i(t)|$}
= Lebesgue constants of interpolation
{c}
{synonym}
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 $Pf=\sum_if(x_i)\ell_i$ 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 $\|f-Pf\|_\infty\le(1+\Lambda_{\mathbf x})\inf_{s\in\operatorname{range}P}\|f-s\|_\infty$.
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