Any nonlinear function $f(x)$, $x \in \mathbb{R}^n$ can be approximated by using orthogonal polynomial approximation \begin{equation} f(x) = \sum_{i=0}^{N_f} b_i q_i(x), \end{equation} where $N_f$ is the maximum order of approximation. $b_i$ are the coefficient, and $q_i(x)$ are orthogonal polynomial, and it satisfies \begin{equation} \begin{split} <q_i(x), q_j(x)> & = 0_{n\times n} \hspace{0.5 cm} I \neq j \\& = I_{n\times n} \hspace{0.5 cm} I = j \end{split} \end{equation}
The coefficient $b_i$ can be calculated as $$b_i = \int f(x) q_i^T(x) dx. $$
My problem is that I did not find more literature on multidimensional polynomial approximation. I just want to know, what will be the dimension of the $b_i$ and $q_i(x)$. I think the $q_i(x)$ will be of dimension $n\times 1$, and the dimension of $b_i$ is $n \times n$. But I don't know, I am doing right or not.
Any suggestion will be great help to me.