Say we have cubic curve $\vec{C}(t)_ = (C_x(t), C_y(t), C_z(t))$ which approximates some parametric function $\vec{F}(t)$ within error less than $\epsilon$. The cubic curve is $C^2$ continuous and is constructed by the usual method of least squares fit to some set of points $(\vec{P}_i, t_i), \quad i<n$ with $\vec{P}_i = \vec{F}(t_i)$. It's the usually called natural cubic spline.
Now, my question is: Can we devise a strategy to reduce the the total number of interpolation points while at the same time ensuring that the approximation error continues to stay bounded by $\epsilon$?
Of course, such strategy should try to use as much as possible of the fact that we start with a approximation which satisfies the error bound.
As a wild idea, I though maybe it would be beneficial to get the control points of each cubic segment, have it in Bezier form so to say, and then try to use convexivity property of the control polygon. Something along those lines. But I am unable to have a bigger picture of the overall algorithm.
An analytical strategy would be gold and would be preferred over one which needs recursive methods. But still recursive strategy would still be better than no strategy.
Thank you very much for your help and answers!