I would like to approximate a continuous (complex) function $f(x)$ in the interval $[a,b]$ $ (x\in\mathbb{R})$ by local polynomial functions of order $3$ (cubic Hermite spline or cubic C2 spline).

Is there an efficient way to find the required number and "optimal" location of sampling points $x_i$, which result in a maximum error $|\overline{f}(x)-f(x)|=\varepsilon$?

I do not know if this question is too general, but the method should work for arbitrary continuous functions f(x). I would also be glad for literature suggestions, which can be handled by an engineer :).


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