This question is related to Master Theorem on oscillating function.
Consider a recurrence of the form
$T(n) = a T(n/b) + f(n)$
Master Theorem's regularity condition excludes some cases (for example, when $f(n)$ is oscillating).
Suppose, however, that $f(n)=\Theta(g(n))$ for a function $g(n)$ that does not violate the regularity condition, so that the Master Theorem is applicable if $g(n)$ is used instead of $f(n)$. Consider then the following recurrence:
$T'(n) = a T'(n/b) + g(n)$
and assume that the master theorem gives the solution $T'(n)=\Theta(g(n))$.
Can I then safely conclude that $T(n)=\Theta(g(n))$? Or there are some reasonable conditions on $g(n)$ we can add (I suppose that if $g(n)$ is polinomially bounded then maybe Akra-Bazzi method will apply, even if one have to swich integration and $\Theta$, and I'm not sure this is sound)?
Notice that from $f(n)=\Theta(g(n))$ we cannot deduce that $f(n)$ is always less than or equal to $g(n)$ or to a $d\cdot g(n)$ for some fixed $d$ (indeed, $g(n)$ can be equal to 0 for some value of $n$, so I'm not totally convinced by the answer to Regularity condition in the master Theorem in the presence of Landau notation for f as it is not possible to prove the base case of the induction, unless one changes the initial condition). So I cannot prove by induction that $T(n)<T'(n)$ for all $n$ and, anyway, I only want to prove $T(n)=\Theta(T'(n))$.
To give some context, this is the case I want to apply the result to: consider the recurrence
$T(n)=2\cdot T(n/2)+f(n)$
where I only know that $f(n)=\Theta(n\sqrt{n})$. Then I can unfold the recurrence obtaining \begin{equation*} \begin{split} T&(n)=2T\left(\frac{n}{2}\right)+\Theta(n\sqrt{n})= 2\left(2T\left(\frac{n}{4}\right)+\Theta\left(\frac{n}{2}\sqrt{\frac{n}{2}}\right)\right)+\Theta(n\sqrt{n}) =\\ &=\ldots=2\left(2\left(2\ldots \left(2T\left(\frac{n}{2^h}\right)+\Theta\left(\frac{n}{2^{h-1}}\sqrt{\frac{n}{2^{h-1}}}\right)\right)\ldots\right)\right)+\Theta(n\sqrt{n}) \end{split} \end{equation*}
until $2^h=n$, i.e. $h=\log(n)$. Then $T(n)=\sum_{i=0}^{\log(n)-1} 2^i\cdot\Theta\left(\frac{n}{2^i}\sqrt{\frac{n}{2^i}}\right)$.
Performing the calculations I get \begin{equation*} \begin{split} T(n)&=\sum_{i=0}^{\log(n)-1} 2^i\cdot\Theta\left(\frac{n}{2^i}\sqrt{\frac{n}{2^i}}\right)=\Theta\left(\sum_{i=0}^{\log(n)-1} 2^i\cdot\frac{n}{2^i}\sqrt{\frac{n}{2^i}}\right)=\\ &=\Theta\left(n\sqrt{n}\cdot\sum_{i=0}^{\log(n)-1} \frac{1}{\sqrt{2^i}}\right) \end{split} \end{equation*} The series $\sum_{i=0}^{+\infty} \frac{1}{\sqrt{2^i}}$ is convergent, so I can conclude (I hope correctly) that $T(n)=\Theta(n\sqrt{n})$.
On the other hand, I cannot directly apply Master Theorem, as from $f(n)=\Theta(n\sqrt{n})$ I cannot conclude $2f\left(\frac{n}{2}\right)<cf(n)$ for some $c<1$. Indeed, if definitively we have $c_1 n\sqrt{n}\leq f(n)\leq c_2 n\sqrt{n}$, then
$f(n)\geq c_1 n\sqrt{n}= 2\sqrt{2}c_1 \frac{n}{2}\sqrt{\frac{n}{2}}\geq\frac{2\sqrt{2}c_1}{c_2} f\left(\frac{n}{2}\right)=\frac{\sqrt{2}c_1}{c_2} 2f\left(\frac{n}{2}\right)$
and so
$2f\left(\frac{n}{2}\right)\leq \frac{c_2}{\sqrt{2}c_1} f(n)$
but $\frac{c_2}{\sqrt{2}c_1}$ is not necessarily less than 1.
