# Choosing basis functions for function approximation

Suppose that $$Y$$ follows multivariate normal $$\mathbb{N}(\mu, \Sigma)$$.

And we know that $$f(Y)$$ follows $$\mathbb{U}(0,1)$$, where $$f: \mathbb{R}^n \to \mathbb{R}$$

Can we find exact form of $$f$$ given this information? I think that we can't, but I can sample pairs $$(Y, f(Y))$$ and use that to approximate $$f$$

Given some basis functions $$\phi_1, \phi_2, ... , \phi_k$$ I could approximate $$f$$ by using least squares:

$$\min\sum (f(Y_i) - a_1\phi_1(Y_i) - a_2\phi_2(Y_i)-... - a_k\phi_k(Y_i))^2$$

My question is: Are there any smart ways to choose basis $$\Phi$$, given that I know distribution of $$Y$$ and $$f(Y)$$