Does a pair of functions $f_1$ and $f_2$ exist between some real interval $[a, b]$ where, given uniformly distributed input x~U[a, b], the Pearson correlation $\rho$ between $f_1(x)$ and $f_2(x)$ may be explicitly controlled as an input to $f_1$ and $f_2$?
To put in another way, does there exist $f_1(x, \rho_0), f_2(x, \rho_0))$ such that $\rho_0=\rho[f_1(x, \rho_0), f_2(x, \rho_0)]$, where $\rho_0 \in [0,1]$ is the input correlation and $\rho$ is the Pearson correlation function,
$\rho(f_1, f_2) = \frac{Corr(f_1[x], f_2[x])}{\sigma(f_1[x])\sigma(f_2[x])}$
where $Cov$ is the covariance function and $\sigma$ is the standard deviation.
The covariance function is defined as
$Cov(f_1, f_2) = \int [f_1 - \int f_1 P(x) dx][f_2 - \int f_2 P(x) dx] dx$
where $P(x) = \frac{1}{b-a}$.
It would be useful for me to have a function where the correlation over $[a, b]$ may be controlled as an input, but I can't think of one. Is it even possible to construct such a function?