# Curve Fitting: Multidimensional input

According to Bishop's book, if your input is a multidimensional vector $x$, then non kernel based basis function models suffer from the curse of dimensionality:

But the above picture is based on polynomial basis functions in one dimension, $\phi_i(x)=x^i$.

If we had different basis functions, like sigmoidal $\phi(x)=\sigma((x-\mu_j)/s)$ where $\sigma(a)=(1/(1+exp(-a))$, would the sum still be equal to the above picture or would the combinations between the different dimensions differ?

For example: You have monomials in $x$, $y$ and $z$ for degree in every direction up to 5. You will than have $3^6$ basis functions (1, $x$, $y$, $z$, $x^2$, ..., $x^5y^5z^4$, $x^5y^5z^5$), by using every chance of combining these to get new basis functions.