What is the name of this type of interpolation/approximation? Piecewise tangent lines' sections / Fabius function example?
Let $f(x)$ be a differentiable function, and let $\{x_i\}$ be a set of equidistant points such have a fixed difference value of $\Delta x=x_i-x_{i-1}$ which is constant for all index $i$.
Then the tangent line to any point of $f(x)$ at some value $x_i$ will be $f'(x_i)(x-x_i)+f(x_i)$, so one could make an interpolation approximation of $f(x)$ by joining these sections of tangent lines: $$T(x) = \sum\limits_i \left( f'(x_i)(x-x_i)+f(x_i) \right)\cdot\frac{\left(\text{sgn}\left(x-x_i+\frac{\Delta x}{2}\right)-\text{sgn}\left(x-x_i-\frac{\Delta x}{2}\right)\right)}{2}$$
I know beforehand it could look something "obvious" or "stupid" since $T(x)$ is not continuous neither differentiable, as other existent interpolation alternatives do are like Splines or Taylor expansions, but I think on it because of the failing of Taylor series in fitting some smooth functions as is explained in the wiki Non-analytic smooth function, as also no Power Series could represent accurately a flat function since matching a constant violates the Identity theorem, but instead $T(x)$ will always converge to $f(x)$ if $f(x)$ is differentiable $|f'(x)|<\infty$, since if $\Delta x\to 0$ then $(x-x_i)|_{x\in\left[x_i-\frac{\Delta x}{2},\ x_i+\frac{\Delta x}{2}\right]}\to 0$ making $T(x)|_{x_i}=f(x_i)\Rightarrow T(x)\overset{\Delta x\to 0}{=} f(x)$.
With this, I could estimate the plot of the differentiable function $f(x)$ by using $T(x)$ if I know some values of $f(x)$ and corresponding values of $f'(x)$: as example, thinking in the case of the Fabius function which "is an example of an infinitely differentiable function that is nowhere analytic", it should be possible to built $T(x)$ since its values at dyadic numbers, if I am not mistaken, in principle should be found as is shown here and here.
So I would like to know:
- The name of this kind of interpolation through tangent lines' secrions in order to look for them and see their properties.
- It is possible to find $T(x)$ for the Fabius function?
Motivation
I am trying to understand if this kind of interpolation is what have been done on the approximation showed in this answer for approximating the Fabius function where the function's values are known only in some points (dense set of points).
Added later
The proposed interpolation looks alike the linear interpolation of two points, but with the difference that here the effective value of the derivative is used instead of its approximation: Are both related? Or equivalent?
I thought at first that the name would be easy to find since it is a really straightforward construction, but misteriously it don't show up on Google: I found this paper by Alan Hurwitz where something really similar is done, but I couldn't figure out if it were exactly the same (my math skills are limited).
Here is a link to Desmos were it can be seen that it kind of fit the function with enough "pieces", even with flat functions where Power Series should have difficulties.
2nd Added later - closing motivation part
By playing in Desmos that $T(x)$ only coincides with the construction done in the mentioned answer at the first order when the fit is done with straight lines: for higher order they don't much being the other method much more inteligent since gives a continuos and differentiable fit, but $T(x)$ still get closer with smaller intervals.
So now at least I know both scheemes are different each other.
