I have a question regarding Lebesgue constant $\Lambda_{n}\left(\boldsymbol{\chi}\right)$, with which the worst case error between an interpolant $p\left(\boldsymbol{x}\right)$ and the function which has to be interpolated, $f\left(\boldsymbol{x}\right)$, can be estimated. In the following $p^{\ast}\left(\boldsymbol{x}\right)$ represents the optimal interpolating polynomial. My question is: Why is always assumed in the definition of the Lebesgue constant, that the funtion $f\left(\boldsymbol{x}\right)$ stems from the space of continuous functions? I don't see, where this has to be used in the derivation of the constant. $$ \left|p\left(\boldsymbol{x}\right)-f\left(\boldsymbol{x}\right)\right|=\left|p\left(\boldsymbol{x}\right)-p^{\ast}\left(\boldsymbol{x}\right)+p^{\ast}\left(\boldsymbol{x}\right)-f\left(\boldsymbol{x}\right) \right| $$ $$ \leq \left|p\left(\boldsymbol{x}\right)-p^{\ast}\left(\boldsymbol{x}\right)\right|+\left|p^{\ast}\left(\boldsymbol{x}\right)-f\left(\boldsymbol{x}\right) \right|\\ $$ $$ =\left|\sum \limits_{l=1}^{n}\left(f\left(\boldsymbol{\chi_{l}}\right)-p^{\ast}\left(\boldsymbol{\chi_{l}}\right)\right)\varsigma_{l}\left(\boldsymbol{x}\right)\right|+\left| p^{\ast}\left(\boldsymbol{x}\right)-f\left(\boldsymbol{x}\right) \right| $$ $$ \leq \sum \limits_{l=1}^{n}\left|\left(f\left(\boldsymbol{\chi_{l}}\right)-p^{\ast}\left(\boldsymbol{\chi_{l}}\right)\right)\left|\cdot \right|\varsigma_{l}\left(\boldsymbol{x}\right)\right|+\left|p^{\ast}\left(\boldsymbol{x}\right)-f\left(\boldsymbol{x}\right) \right| $$ $$ \leq \left\|\left(p^{\ast}\left(\boldsymbol{x}\right)-f\left(\boldsymbol{x}\right)\right)\right\|_{\infty} \sum \limits_{l=1}^{n}\left|\varsigma_{l}\left(\boldsymbol{x}\right)\right|+\left|p^{\ast}\left(\boldsymbol{x}\right)-f\left(\boldsymbol{x}\right) \right| $$ $$ \leq \left\|\left(p^{\ast}\left(\boldsymbol{x}\right)-f\left(\boldsymbol{x}\right)\right)\right\|_{\infty} \left\|\sum \limits_{l=1}^{n}\left|\varsigma_{l}\left(\boldsymbol{x}\right)\right|\right\|_{\infty}+\left\|p^{\ast}\left(\boldsymbol{x}\right)-f\left(\boldsymbol{x}\right) \right\|_{\infty} $$ $$ =\left(1+ \Lambda_{n}\left(\boldsymbol{\chi}\right)\right)\left\|p^{\ast}\left(\boldsymbol{x}\right)-f\left(\boldsymbol{x}\right) \right\|_{\infty}. $$ And therefore $$ \left\|p\left(\boldsymbol{x}\right)-f\left(\boldsymbol{x}\right)\right\|_{\infty} \leq \left(1+ \Lambda_{n}\left(\boldsymbol{\chi}\right)\right)\left\|p^{\ast}\left(\boldsymbol{x}\right)-f\left(\boldsymbol{x}\right) \right\|_{\infty}. $$ In the thrid row I have made use of the fact, dass every interpolating polynomial of degree < n is its own interpolant of the same degree. Furthermore I have used, that the Lagagrange interpolant $p\left(\boldsymbol{x}\right)$ is equal to $\sum\limits_{l=1}^{n}f\left(\boldsymbol{\chi_{l}}\right)\varsigma_{l}\left(\boldsymbol{x}\right)$. The $\varsigma_{l}\left(\boldsymbol{x}\right)$ represent the Lagarange polynomials.
All the best
Dominik/Germany