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.


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