Differentiability of functions in reproducing kernel Hilbert space

Question

Consider $\mathcal H_k$ to be the RKHS of a reproducing kernel $k$. I am interested in the differentiability properties of $\mathcal H_k$ as a space of functions.

More precisely, is there a link between the differentiability properties of $k$ and that of the functions in $\mathcal H_k$ ?

In the Gaussian process litterature, there are results about mean square differentiability of the sample paths of Gaussian processes associated to the covariance function $k$. For instance if $k$ is defined on the real line, mean square differentiability of the sample paths amounts to continuous differentiability of $k$. Yet this does not inform us on the properties of the functions in the RKHS since sample paths do not belong to the RKHS in general.

Does anyone know of results on differentiability of the functions in RKHS ? I am especially interested in the case of the Matérn kernel, which is supposed to result in RKHSs of more or less smooth functions depending on the parameter values. But what does "smooth" mean exactly ?

Answer 1

I have found an answer in the book Wendland (2004). From their theorem 10.45 (which deals with the more general case of conditionally positive definite reproducing kernels of which positive definite reproducing kernels are a particular case) we have that

Let $\mathcal X$ be an open subet of $\mathbb R^q$, and let $k$ be a positive definite reproducing kernel on $\Theta$ which is $2s$ times continuously differentiable on $\mathcal X$ ($k \in \mathcal C^{2s}(\mathcal X \times \mathcal X)$, then its associated RKHS $H_k$ is a subset of $\mathcal C^{s}(\mathcal X)$ (i.e. elements of the RKHS are then $s$ times continuously differentiable on $\mathcal X$).

Additionally for the Matérn kernel, a link can be made with fractional Sobolev spaces. More precisely, the Matérn kernel is defined as $$ k_{\nu, h}(x, x') = \frac {2^{1 - \nu}}{\Gamma(\nu)} \Big( \frac {\sqrt {2 \nu} \| x - x' \|} h \Big)^\nu B_{\nu} \Big( \frac {\sqrt {2 \nu} \| x - x' \|} h \Big)~, $$

Then we have the following theorem (Corollary A.6 in Tuo and Wu (2016))

For $ \lfloor \nu + \frac q 2 \rfloor > \frac q 2$, the RKHS generated by the Matérn kernel equals the (fractional) Sobolev space of order ${\nu + \frac q 2}$ with equivalent norms.

Where we have denoted by $\lfloor a \rfloor$ the integer part of $a \in \mathbb R$.