Let "$k$-compression" ($k\in\mathbb Z, k\geq 2$) of formal power series be a map $\mathcal C_k : R[[X]] \to R[[X]]$ such that $\sum_{n=0}^{\infty} a_n X^n \mapsto \sum_{n=0}^{\infty} a_{n \cdot k} X^n$.
Does a closed-form for $\mathcal C_k$ exist in general? Does it exist in some special cases?
I have found the following special case:
Let $X$ be $\mathbb R$, then for $k = 2$ $$\frac{A(\sqrt x) + A(-\sqrt x)}{2} = \sum_{n=0}^{\infty} a_{2n} x^n$$
This is obviously hand-wavy because $\sqrt x$ isn't a power series, therefore its' composition with $A$ isn't formally defined.
This generalizes to $k\in\mathbb Z, k\geq 2$ using the second answer to this question and composing it with $\sqrt[n]{x}$.