I compute (at a great cost) upper and lower bounds $f_u(x)$ and $f_l(x)$ of an unknown function $f(x)$ at points $x$ in $[0,1]$. Now I am interested in an estimation of the derivative $f'(x)$. I know a theoretical upper bound for the second derivative of $f(x)$, i.e. I know a $K$ so that $|f''(x)| < K$ in the whole interval. Is there are standard method for generating an estimate of $f'(x)$ in this case?
I am currently fitting my data to Chebyshev polynomials up to some order, and then I take the derivative of the fitted function, but this does tell me how big the error might be, and it does not generate the smoothest possible $f(x)$ that respect the two known bounds.