For $f(x)=\sum_{n=0}^\infty a_nx^n$ and $g(x)=f(x)-Rf(x^2)$, $R\neq 1$, the series $\sum_{n=0}^\infty a_n$ belongs to the elementary Ramanujan class $R$ if $g(x)$ converges at least for $|x|<1$ and there exists $g(1)=\lim_{x\to 1^-} g(x)$. The elementary Ramanujan sum of $\sum_{n=0}^\infty a_n$ is defined by $f(1)=g(1)/(1-R)$. The Ramanujan class $R$ is a linear space that includes the linear subspace of Abel summable series $\mathcal{A}$ (the Ramanujan class $R=0$) (definition, also here).
The extension of the elementary Ramanujan summation to the direct sum of all Ramanujan classes $\mathcal{R}$ is well-defined (here). Is $\mathcal{R}$ closed under iteration?
For $f_k(x)=\sum_{n=0}^\infty a_{kn}x^n$ belonging to distinct Ramanujan classes $R_k$, $k=1,\cdots ,N$, at $x=1$, and $g(x)=f_1(x)+\cdots + f_N(x)$ (instead of $g(x)=f_0(x)$), does $\sum_{n=0}^\infty a_n$ belong to $\mathcal{R}$?
Edit 1.
$f(x)$ is determined by $g(x)$ and $R$: $a_0=\sum_{k=1}^N a_{k0}/(1-R)$, $a_{2n+1}=\sum_{k=1}^N a_{k 2n+1}$, $a_{2(n+1)}=Ra_{(n+1)} + \sum_{k=1}^Na_{k 2(n+1)}$.
For $N=1$, $R\neq R_1$ and $f_{1R}(x)$ defined by $$ f_{1R}(x)-Rf_{1R}(x^2)=-\frac{R}{R_1-R}g_1(x), $$ where $g_1(x)=f_1(x)-R_1f_1(x^2)$, we obtain $$ f(x)=\frac{R_1}{R_1-R}f_1(x)+f_{1R}(x)~. $$ Thus, $\sum_{n=0}^\infty a_n$ actually belongs to $\mathcal{R}$.
Edit 2.
As $f_1(1)= g_1(1)/(1-R_1)$ and $f_{1R}(1)=-Rg_1(1)/(R_1-R)(1-R)$, we verify that $f(1)=R_1f_1(1)/(R_1-R) + f_{1R}(1)=f_1(1)/(1-R)$.
Linearity implies the general result for $R\neq R_k$, $k=1,\cdots ,N$.
Edit 3.
However, for $R=R_k$, $\sum_{n=0}^\infty a_n$ does not belong to $\mathcal{R}$. For example, $f(x)-2f(x^2)=x/(1-x)$ corresponds to the series $$ 1+3+1+7+\cdots=\frac{1}{2} $$ under iterated elementary Ramanujan summation.
No. Iteration generates a hierarchy of elementary Ramanujan summation methods.
Consider a sequence $f_k(x)=\sum_{n=0}^\infty a_{kn}x^n$, $k=1,2,3,\cdots$, such that $f_k(x)-Rf_k(x^2)=f_{k-1}(x)$, $R\neq 1$, and $f_0(x)$ is Abel summable at $x=1$. The iterated elementary Ramanujan sum of $f_k(x)$ at $x=1$ is defined by $f_k(1)=f_{k-1}(1)/(1-R)=f_0(1)/(1-R)^k$. The iterated Ramanujan class $[R,k]$ is the set of all such series $f_0(x), f_1(x),\cdots ,f_k(x)$ at $x=1$.
Edit. $[R,k]$ is a linear subspace. The iterated elementary Ramanujan summation is a linear function defined on the direct sum of all $[R,k]$ ($R$ is a real number, $R\neq 1$, and $k$ is a natural number, $k\neq 0$).