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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.

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1 Answer 1

If you know that $\|f''\|_\infty<K$ then you should have a bound on $\|f'\|_\infty$ in the same way that a bound on $\|f'\|_\infty$ leads to a bound on $\|f\|_\infty$ (given some information on $f$ -- so we will look at this).

I am assuming that $f$ is differentiable on $[0,1]$ (twice differentiable on $[0,1]$ in the sequel). You can show that if $\|f'\|_\infty<L$ then $f(a)+L$ is an upperbound for $f$ on $[0,1]$ (for a proof, suppose that $f(x)>f(0)+L$ for some $x\in(0,1]$ and show that this contradicts the Mean Value Theorem).

We can probably adapt this to your problem where $f'$ plays the role of $f$ and $f''$ the role of $f'$. Using it straight away we have

$$\|f'\|_\infty\leq f'(0)+K.$$

Now we don't know $f'(0)$ but I suppose but we could just use any $f'(x)$ for $x\in(0,1)$ if we got one. We can do this using the Mean Value Theorem if we know that $f(a)=f_u(x)$ and $f(b)=f_l(x)$ (or we could probably do with estimates --- let's run with estimates).

Suppose that we have $m\leq f(x)\leq M$ (just take $m=f_l(x)$ and $M=\max\{|f_u(x)|,|f_l(x)|\}$.)

Now, can we say anything about

$$\frac{M-m}{x_M-x_m},$$

where $x_M$, $x_m$ are points that are 'close' to the maxima: $f(x_m)\sim f_m(x)$ and $f(x_M)\sim f_M(x)$. Well we have a problem if the max and min are close together...

I'm stuck at this point but I hope this hint helps --- any more information on the problem (how far apart are the maxima/ minima found, is there a local max/ min inside $[0,1]$ so that we could take $f'(a)=0$ there and work from this? Two pieces of information about $f$ or two about $f'$ will do.) and we might be able to take these problems to fruition.

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