we consider the following dynamics \begin{equation}\label{dotxt} \dot{x}_T = g(x_T) + y, \end{equation} where $g(x_T)$ is a time-varying continuous function given by \begin{equation}\label{g} g(x_T) = \alpha_1\left(x_T-y\right)\mathrm{e}^{-{\lVert x_T-y\rVert}^2} + w_T -y. \end{equation} and the disturbance term $w_T$ and input $y$ is what we already know, then $g$ is a function with the variable $x_T$.
We let $y$ as the control input given by \begin{align}\label{input} y &= -kx_T-\sum_{i=1}^{m}\hat{d}_i\psi_i, \\ \dot{\hat{d}}_i &= x_T\psi_i, \end{align} where $m$ is a positive constant and $\psi_i$ is the $i$th Gaussian function and $d_i$ is the $i$th unknown constant.
From Stone–Weierstrass theorem, we use $m$ Gaussian functions to approximate the continuous function $g(x_T)$. Let \begin{equation} f(x_T) = \sum_{i=1}^m d_i\psi_i(x_T) + \epsilon, \end{equation} where $\epsilon$ is the approximation error. then \begin{equation} \dot x_T = y+ \sum_{i=1}^m d_i\psi_i(x_T) + \epsilon. \end{equation}
My question is, for using finite numbers of Gaussian or other basic functions to make the approximation, is there any bound for the approximation error? Is there any theorem that solves the order of magnitudes of the $\epsilon$?