First, I would like to mention that what you define as 'error' is pretty important. You could obtain error estimates based on an assortment of different norms or error measures, each most likely resulting in a slightly different solution. Optimality will always be depending on the error/cost function.
Next, I personally formulated it as a Weighted Nonlinear Least Squares problem, where the location of each x point used for interpolation is a part of a vector seeking to be optimized using the standard Weighted Least Squares styled cost function.
I did a first attempt trying to use gradient descent with this formulation, trying out the results with a few different guesses. A guess I formulated before I even attempted this numerically has found to be roughly optimal when I plugged it into the numerical algorithm.
This sub-optimal analytical solution is:
$$x = e^{(-as)}$$
where $a = -ln(2048)$, and $s \in [0, 1]$. For this problem, choose 10 equally spaced apart values for $s$ starting at $s$ = 0 and ending at $s$ = 1. This generates a set of x values increasingly spreading apart as x increases.
Now, I got some pretty worthwhile results with the approach I did, but I am not sure if it's totally optimal or not with respect to the cost function I made. I would recommend tackling this problem using a more global optimization approach, such as Particle Swarm Optimization, Genetic Optimization, etc.
Here's some possibly helpful references:
Basic $L^p$ Norm Info
Least Squares Info
