Recall the definition of Hardy's hierarchy:


$H\_0(n)=n+1\\
H\_{\alpha+1}(n)=H\_\alpha(n+1)\\
H\_\alpha(n)=H\_{\alpha[n]}(n)$, 


where the last rule applies if $\alpha$ is limit ordinal, and $\alpha[n]$ for $\alpha<\varepsilon\_0$ are fundamental sequences (we are using their standard definitions). We can generalize this definition to start with any other increasing function $F(n)$:


$F\_0(n)=F(n)\\
F\_{\alpha+1}(n)=F\_\alpha(n+1)\\
F\_\alpha(n)=F\_{\alpha[n]}(n)$


This hierarchy has the property that for each $\alpha<\beta$ we have that $F\_\alpha$ is dominated by $F\_\beta$.


What I'm interested in are functions which, in a way, lie between $H\_\alpha$ for $\alpha<\varepsilon\_0$ and $H\_{\varepsilon\_0}$ itself. To be more specific, I'm looking for a function $F$ which dominates each of $H\_\alpha$, and $H\_{\varepsilon\_0}$ dominates each of $F\_\alpha$, for $\alpha<\varepsilon\_0$ (similar question for $\varepsilon\_0$ replaced with smaller ordinals). I don't know if any functions of the type described above exist, and my question is: are such functions known to exist? I suspect something of this sort could've been investigated by people such as Andreas Weiermann, but I haven't found anything of my interest.


Thanks in advance.



Functions in "gaps" in Hardy hierarchy
The function F exists. This follows from results from subrecursive degree theory.
A possible reference is: Streamlined subrecursive degree theory. 
Lars Kristiansen, Jan-Christoph Schlage-Puchta & Andreas Weiermann
Annals of Pure and Applied Logic 163 (6):698-716 (2012)


Andreas Weiermann


