Suppose I have a function $f \in \mathcal C^{\alpha, L}([0,1])$, where $\mathcal C^{\alpha, L}([0,1])$ is the space of $\alpha$-smooth Hölder functions with norm $L$. I am interested in efficiently approximating $f$ using a kernel-based method; we might have \begin{align*} \widetilde f(x) = \sum_{j = 1}^k \mu_j \psi\left( \frac{x - c_j}{\sigma} \right), \end{align*} where $\psi(x)$ is something like a Gaussian. I am particularly interested in a logistic kernel $\psi(x) = e^{-x}(1 + e^{-x})^{-2}$.
I am reading a statistics paper which makes the following claim
Approximation theory tells us that for a compactly supported kernel and compactly supported $\beta$-Hölder function, being not necessarily nonnegative, the approximation error will be of order $k^{-\beta}$, provided $\sigma \sim k^{-1}$ and the weights are carefully chosen. This remains the case if both the kernel and the function to be approximated have exponential tails, as we consider in this work.
The paper itself is dealing with a density estimation problem, and concerns only power exponential kernels $\psi(x) \propto e^{-|x|^p}$.
I'm in the process of going through a textbook treatment referenced in the paper as a source for this claim (Constructive Approximation by DeVore and Lorentz), but I am wondering if anyone can provide a simple reference which will give me essentially the fact cited above, or even just with application to my logistic kernel. Also, in case it's not clear, I know basically nothing about approximation theory. Extensions of this result to $\mathcal{C}^{\alpha, L}([0,1]^d)$ using a tensor-product kernel $\psi^{(d)}(x) = \prod_{i = 1}^d \psi(x_i)$ would also be very helpful.
