I have uniformly-distributed random variables $x_i$ satisfying
$$0\le x_i\le 1$$
I take a weighted mean of these using weights $w_i$ satisfying
$$0\le w_i\le 1$$ $$\sum w_i=1$$
The naive weighted mean is
$$\hat{x}=\sum w_ix_i$$
but $\hat{x}$ no longer has a uniform distribution.
What smooth adjustment $X=f(\hat{x})$ can I make such that $X$ has uniform distribution?