I've been working on a programming project where the most immediate solution to a problem is to fit a quadratic to the data. However, this doesn't completely work for the data since sometimes there are regions which essentially have increased or decreased slopes than what would be expected from a standard quadratic. An example of such a function one would like to fit can be seen in the following figure:
(Note: I quickly made this plot as a composition of 3 quadratic functions with differing slopes, what would've been more optimal representation would be if there were no kinks at $x=-2$ and $2$)
Although one can not to difficultly solve for individual cases, this problem motivated a more general question of mine which I cannot think up a solution for.
Does there exist a smooth function $f: \mathbb R \rightarrow \mathbb R$ such that:
- $f^{(n)}=0 \ \forall n>k$
- $f^{(i)}> b_i, \ b_i \in \mathbb R$ $\forall i \in I, \ I \subset \{1, \dots, k\}$
- $f \neq \sum_{j=0}^{k}a_jx^j$
Edit: I overlooked the fact that 1 & 3 together imply a contradiction. Thus I'd like to alter my question to finding examples of functions where 2 & 3 are at play.
Reference recommendations, examples or a description of such a function in its generality would all be greatly appreciated.
No there can't be, because your first and third assumption do not work together.
If you derive a function once and get zero then the function must have been constant. If you do it twice, you get that the function had to have been a linear function and so on. So you will always get a polynomial.
Moreover your plot seems to have a kink in it, so searching for a n times differentiale function to fit this, might not be a good idea.