# Which frequency bins give the best interpolation for the derivative of a function?

A function $$u:[0,2\pi]\to\mathbb R$$ sampled over $$N$$ equidistant points $$\theta_j=(2\pi/N)j,\, j = 0, \dots, N-1,$$ can be interpolated by a set of functions $$\{u_{k_0}\}$$ enumerated by integers $$k_0\in\mathbb Z$$. $$u_{k_0}(\theta) = \sum_{k=k_0}^{k_0+N-1} \tilde u_k\, e^{ik\theta}\,, \quad \text{where}\quad \tilde u_k = \frac1N \sum_{j=0}^{N-1} u(\theta_j)\, e^{-ik\theta_j}\,.$$

This gives us an infinite sequence of interpolations for the derivative of the above function: $$u'_{k_0}(\theta) = \sum_{k=k_0}^{k_0+N-1} ik\, \tilde u_k\, e^{ik\theta}\,, \quad k_0 \in \mathbb Z.$$

I would like to know if there is any value of $$k_0$$ for which the corresponding interpolation gives the best estimate for the derivative. I have a feeling that it is $$k_0 = -N/2$$.

Kindly provide your argument for why such a special $$k_0$$ exists if it does. Thank you.

P.S. To give you context, please know that I have no prior experience in signal processing. In related questions, I have heard a lot of talk about "Nyquist frequency", "band-limitation", etc. which concepts I am not very familiar with.