Let $F = \{F_1, F_2, \dots, F_n\}$ be a set of functions of the form $F_i: S \to \mathbb R$. Additionally, let $X$ be a random variable with state space $S$.
The average correlation of $f: S \to \mathbb R$ (not necessarily in $F$) is defined as
$$\overline{\rho_{f(X), F(X)}} = \frac 1n \sum_{i=1}^n\rho_{f(X), F_i(X)}$$
if $f(X)$ is defined.
My question is is there is function $f: S \to \mathbb R$ with maximal average correlation (by which I mean $f(X)$ is defined and there does not exist a $f': S \to \mathbb R$ such that $f'(X)$ is defined and $\overline{\rho_{f'(X), F(X)}} > \overline{\rho_{f(X), F(X)}}$)?
Since $\overline{\rho_{f(X), F(x)}}$ is bounded above by $1$, a supremum exists. I do not know if a $f$ exists achieving this supermum, though (hence my question).
Also, $f$ is not unique, since if $f$ has maximal average correlation so does $g$ if $g(x) = a + bf(x)$ for $a \in \mathbb R_+$, $b \in \mathbb R_{\neq 0}$, and $x$ ranging over $S$. That said $f$ is likely unique up to this transformation on the support of $X$.
Note also that simply taking the average of $F$ likely will not work, since the pearson coefficient ignores constant factors, whereas the average does not.