I want to compute the greatest $a>0$ for given $\epsilon>0$ such that $$max_{x\in [-a,a]}|f(x)-p_2(x)| < \epsilon$$ where $a$ is the distance between two grid points and the maximum is the interpolation error. $p_2$ is the interpolation polynomial of $f(x)=sin(2x)$ with $p_2(x)=\frac{sin(a)}{a}x$
We know that $$max_{x\in [0,a]}|f(x)-p_2(x)| <\omega_n(x) max_{\xi\in[0;a]} \frac{f^{n+1}(\xi)}{(n+1)!}$$ With $\omega(x)$ being knot polynom.
Because as JeanMarie mentioned we just look at the inervall $[0,a]$, I got: $\omega(x)=x(x-a)=x^2-xa$
$max_{\xi\in[0;h]} \frac{f^{n+1}(\xi)}{(n+1)!}$=$max_{\xi\in[0;a]} \frac{4}{3}cos(2\xi)$ $\Rightarrow$ Maximum in $\xi=0$
Now shall I take an arbitrary $\epsilon>0$ and find the greatest $a>0$ ?