Characterising a function entailing limit and also minutes of various other features

Given $n$ smooth actual features $f_1, f_2, \dots, f_n$, specify a composite function similar to this:

$$f(x) = \max(f_1(x), f_2(x), \dots, f_n(x)) - \min(f_1(x), f_2(x), \dots, f_n(x))$$

Is it feasible to claim anything valuable as a whole concerning the form of this function?

With ease, it feels like $f$ will certainly go to the very least $C^0$ continual yet $f^\prime$ might have randomly several stoppages. Just how much would certainly we need to find out about the specific $f_k$ to be extra details?

As an example, if we understand that each $f_k$ has $m_k$ extrema, it feels like we need to have the ability to position bounds on both the variety of extrema in $f$ and also variety of stoppages in $f^\prime$, yet I'm having problem thinking of all the feasible communications as $n$ rises.

(I've attempted to place this as a whole terms, but also for context my certain passion is rather pertaining to my earlier optics question. A various yet comparable imaging procedure generates something like the above function with basic $f_k$ of this kind:

$$f_k(x) = \sum_i e_i \sin(x_i + \frac{2k\pi}{n}) P(x - x_i)$$

Once once more, simulations recommend that this procedure can visibly boost side resolution as a result of the edges presented in between optimums, yet it would certainly behave to have an extra official characterisation.)

2019-05-18 20:51:25
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Answers: 1

Everything you require to recognize can be reasoned from the identification $\text{max}(a, b) = \frac{a+b}{2} + \frac{|a-b|}{2}$ and also in a similar way for $\text{min}$.

Edit: Okay, possibly this is much less valuable than I assumed. You need to know concerning the extrema of $f$ and also the stoppages of $f'$. If each $f_k$ has finitely several extrema, it might still hold true that $f$ has definitely several stoppages ; take into consideration 2 features which are raising in a "race" in which the victor adjustments definitely usually. As a removed solution claims, you desire on top of that that $f_i - f_j$ has finitely several extrema for every single set $i, j$. After that outside a bounded series of values of $x$ the loved one order of the $f_i$ will not transform and also every little thing will certainly be hunky - dorky.

2019-05-21 02:17:14