Let $f:[0,1]^n\to (0,\infty)$ be a continuous function. The stone-weierstrass theorem gives $\forall \epsilon >0$ we have a polynomial $p$ s.t $||f-p||_{\infty}<\epsilon$. I was wondering if we can approximate $f$ with a non-linear functions of the following form: $$p=\sum_{j=1}^k\alpha_j\prod_{i=1}^nx_i^{\beta_{i,j}}$$ where $\alpha_i,\beta_{i,j}\in \mathbb{R}$ s.t $\alpha_i>0$ and $\beta_{i,j}>0.$
Is this possible?
No. Your polynomials of interest are those with all positive coefficients, but such functions are monotonic increasing on the unit interval. Hence they cannot approximate e.g. a decreasing function.