I have a smooth function :
$$\frac{1}{2\pi} \sin(2\pi x )~,$$
that I would like to approximate on an interval $[-a, b]$. However I already know that all the inputs are guarantied to be close to an integer :
$x = i + \epsilon$, where $-a \leq i \leq b \in \mathbb{Z}$ and $-2^{-e} \leq \epsilon \leq 2^{-e}$, for some given $e$.
So I do not need to approximate the function over the whole interval, but only near the area where my inputs will fall.
Is there a way to compute the Chebyshev nodes in such a way that I can concentrate them near the expected input values in order to reduce the approximation degree for an equivalent precision? If yes, how would that be done?