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Given a function f and an unique interpolation polynomial P, we can say that for every x there is a r so that

$f(x)-P(x)=\frac{\omega(x)f^{n+1}(r)}{(n+1)!}$

where r is in the smallest interval $[x_0,...x_n,x]$ that contains x and all support points of P.

Now I need to show that for $h:=max_{i=0,n-1} (x_{i+1}-x_{i})$

we can estimate $\omega$ as

$max_{x\in[x_0,x_n]}|\omega(x)| \leq \frac{n!}{4}h^{n+1})$

We know that $\omega =(x-x_0)(x-x_1)...(x-x_n)$.

I tried to transform the very first formula because the estimation contains a factorial but unfortunately I don't really see how I can show this estimation.

illuminatitruthseeker
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  • One thing I just noted is that $max_{x∈[x_0,x_n]}|ω(x)|$ must be 0, right? Because if I plug in any $x_i$ in ω(x), one of the linear factors becomes 0 so the whole left side becomes 0. So do I only need to show that the right side is >= 0 or am I completelty wrong on this attempt? – illuminatitruthseeker Nov 23 '21 at 14:35
  • How do "we know that $\omega = (x - x_0)(x-x_1)...(x - x_n)$"? I do not see why that must hold at all. Indeed, I believe that contradicts the estimate you are trying to prove. I think you may have mixed up two different functions that were labelled $\omega$ in your text. – Paul Sinclair Nov 24 '21 at 00:30

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