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](http://projecteuclid.org/download/pdfview_1/euclid.ejs/1289226500) 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. 



Constructive approximation of Hölder functions using kernel functions
If you are just concerned with logistic kernels, and you are willing to put a mild assumption on the $\beta$-Hölder function $f$ to be estimated, then [Rousseau] assumes mild condition $\boldsymbol{A}\_0$ and proved the $k^{-\beta}$ decay in Theorem 3.1, which is also cited by [Kruijer&Rousseau]. The good thing is that with this additional assumption the resulting approximation is explicitly given there. The downside is that a more general result in [DeVore&Lorentz] 2.6~2.7 holds.


[DeVore&Lorentz] discussed this thoroughly around Chap 2. As for tensor kernel, a popular easy-catch is [Gu]'s books on splines, which basically eliminated all theoretic complications but still point to useful results in statistics.


BTW, sometimes you need to swallow down a bit math, statistics is no more than a variant of mathematics:-)


**Reference**


[Kruijer&Rousseau]Kruijer, Willem, Judith Rousseau, and Aad Van Der Vaart. "Adaptive Bayesian density estimation with location-scale mixtures." Electronic Journal of Statistics 4 (2010): 1225-1257.


[Rousseau]Rousseau, Judith. "Rates of convergence for the posterior distributions of mixtures of betas and adaptive nonparametric estimation of the density." The Annals of Statistics 38.1 (2010): 146-180. <https://projecteuclid.org/download/pdfview_1/euclid.aos/1262271612>


[Gu]Gu, Chong. Smoothing spline ANOVA models. Vol. 297. Springer Science & Business Media, 2013.


