I have a number which is made up of a Harmonic series. 1/2 + 1/3 + 1/4 etc. Some of the components may not be in the number.. 1/2 + 1/7 + 1/11 etc.
Is it possible to recover the individual components? Is this basically a FFT of some sort?
In my case my ultimate goal is to continue the decay.. So 1/2 + 1/7 + 1/11 steps to 1/3 + 1/8 + 1/12.
In general this is a no, because multiple sequences can generate the same number. If I have something as simple as
$$\frac 1x+\frac 1y=\frac ab$$
(so this "harmonic sequence" has all but two elements removed), I have multiple solutions.
Take $a/b=1/6$. Then $x=8,y=24$ is one solution and $x=7,y=42$ is another. So if you say "I've added up several fractions of the form $1/x$ to produce the answer $1/6$. Which fractions did I pick?" The question doesn't have a unique answer, so the number doesn't have a particular "fourier transform". Even for the simple case of
$$1/x+1/y=1/z$$
The number of solutions (counting $x,y$ swapped as a distinct solutions, for simplicity) is the number of factors of $z^2$. Allowing the right fraction to be arbitrary, or increasing the number of terms, just brings in more potential solutions.