This is a follow-up question to this one where I asked if asymptotic equivalence of coefficients carried over after convolution, resp. why this was not the case. Answerer Daniel Fischer proposed that a weaker form may hold under certain conditions. This question is to explore this further and provide some information about the connection to my original problem, which has since become clearer.
Let $f$, $\tilde f$ and $g$ ordinary generating functions with the following properties.
- $f$, $\tilde f$ and $g$ have the same radius of convergence $0 < \rho < 1$.
- $[z^n]f(z)$ and $[z^n]g(z)$ are positive integers and non-decreasing.
- $[z^n]f(z) \sim [z^n]\tilde f(z)$ for $n \to \infty$.
The question is if
$\qquad\displaystyle [z^n](f \ast g)(z) \overset{?}{\in} \Theta\bigl( [z^n](\tilde f \ast g)(z) \bigr)$,
that is if the coefficients of the (discrete) convolution remain within $\Theta$-bounds if we exchange $f$ for $\tilde f$.
Background
I know for some class of formal languages that for every $\alpha$ in some subset of dyadic numbers, there is a language $L$ with
$\qquad |L_n| \sim c \cdot d^n \cdot n^{\alpha}$
where $c$ and $d$ are suitable constants. While I do not know the exact generating function $F_L$ of such a language -- or don't want to bother with it technically -- I can come up with
$\qquad F(z) = 1 \pm (1-dz)^{-\alpha-1}$
which has the same asymptotic behaviour in coefficients (up to factor $c$ which I happily ignore). Now I take the Cartesian with some fixed language $L_B$ and ask for
$\qquad |(L \times L_B)_n| \in \Theta(?)$.
If I can use $F$ instead of $F_L$ and still get the same $\Theta$ (see above), I can answer this question (more easily).