# Hello. How to prove theorem about relationship between the interpolation operator norm and the constant of Lebesgue:

A norm of the linear operator $$P_n$$ can be defined by

$$||P_n|| = max_{f \in [a,b]} \frac{||P_n f||}{||f||}$$

where on the right-hand side one takes any convenient norm for functions. Taking the $$L_\infty$$ norm, one obtains form Lagrange's formula

$$||p_n (f; \cdot)||_\infty = max_{a \leq x \leq b} |\sum_{i=0}^{n} f(x_i) l_i(x)| \leq ||f||_{\infty} max_{a \leq x \leq b} \sum_{i=0}^{n} |l_i(x)|.$$ Indeed, equality holds for some continuous function f:

Theorem:

$$||P_n||_\infty = \Lambda_n,$$

where

$$\Lambda_n = ||\lambda_n||_\infty, \; \; \lambda_n(x)= \sum_{i=0}^{n} |l_i(x)|.$$

How to prove this theorem?

• What is the connection between $P_n$ and $l_i$'s? I don't see the definition of $P_n$ anywhere. – Kavi Rama Murthy Apr 12 at 11:51