I have the following equation: $ h(n) = n \sum_{i=0}^{\lceil \log_2 n \rceil} \frac{m(2^i)}{2^i} $
and I'm trying to understand exactly the relationship between the functions $h$ and $m$. The easiest way for me to do this is to fix $m$ to some specific function, then try to figure out what $h$ is. For example:
\begin{align} m(n) &= 1 &\Rightarrow h(n) &= n \\ m(n) &= \log n &\Rightarrow h(n) &= n\\ m(n) &= n &\Rightarrow h(n) &= n \log n \\ m(n) &= n^2 &\Rightarrow h(n) &= 2^n \end{align}
It would be really helpful, however, to be able to fix $h$ and find the cooresponding value of $m$. For example, what is $m$ if $h(n)=n^2$?
Is there some general procedure for doing this? Am I even approaching this problem in the right way?
Edit: In particular, I'd especially like to find out what $h$ and $m$ are equal to when $h(n) = m(n)$.