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I'm practising for my exam and here is a problem that I have difficulties to solve:

For a given sequence of different nodes $\left\{x_k \right\}_{k=0}^{n} \subset [a,b]$, let $\left\{l_k\right\}_{k=0}^n$ denotes Lagrange basis polynomials and $L_nf$ interpolation polynomial for $f\in C([a,b])$ in these nodes. Prove that: $$\|L_nf\|_{\infty, \ [a,b]} \le \left( \sum_{k=0}^{n}\|l_k\|_{\infty, \ [a,b]} \right) \|f\|_{\infty, \ [a,b]}$$

Seems very, very hard. Can anyone help?

xan
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1 Answers1

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This follows from $$|L_nf(x)| =|\sum_{k=0}^nf(x_k)l_k(x)| \leq \sum_{k=0}^n|f(x_k)|\,|l_k(x)|.$$

WimC
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