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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?

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

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