Is there any function $f(x)$ which satisfies the following criterion for positive values of $x$ :
- being positive ($f(x)>0$)
- having negative odd derivatives ($f^{2k+1}(x)<0$)
- having positive even derivatives ($f^{2k}(x)\gt0$)
The exponential function $f:x\rightarrow e^{-x}$ does satisfy such conditions but I would like to have a function that could be used to fit some experimental data. Therefore, I want $f(x)$ to have at least 3 parameters that can be adjusted and which optimal values over the experimental data will ensure the previously mentioned inequalities.
For example, $f:x\rightarrow x^{c_1}\exp \left[c_2 (1-x^{c_3})\right]$ has 3 parameters $c_i$ but does not satisfy the constraints previously mentioned for all values of $c_i$ and therefore is not good for me.
Any idea ?
This is not a mathematics question at heart, but it has a certain overlap with mathematics.
Such functions are called completely monotone, and are characterized by being the Laplace transforms of non-negative distributions. (Look for Bernstein's theorem.)
So you could regard your experimental data as measuring the Laplace transform of some unknown non-negative function or measure which you model somehow. For instance, as a mixture of delta functions with weights and locations you fit, or as a histogram whose bar heights you fit, or as the exponential of a polynomial whose coefficients you fit: something like that. How you actually do this is certainly not a mathematics problem but rather a question of what makes sense in your application area. Expect a certain amount of computer huffing and puffing.