RKHS norm and $L_\infty$ norm under Gaussian kernel

Question

Given a gaussian kernel $k(x) = \exp(-||x||^2/2)$, let $H_k$ be the associated Reproducing Kernel Hilbert Space(RKHS). My question would be: is it possible to obtain an upper bound of $||f||_{H_k}$ in terms of $||f||_{L_\infty}$ for any $f \in H_k$? The other direction seems to be easy.

Answer 1

Under no further assumptions on $f$, I'd say that such an upper bound is unlikely. For instance, the Gaussian kernel's RKHS does not contain constants, other than the zero function (see Corollary 4.44 in Steinwart & Christmann, 2008, for a proof). Then any function approaching a constant $c > 0$ everywhere would have $\lVert f \rVert_{L_\infty} \to c$, but $\lVert f \rVert_{H_k} \to \infty$. It might be possible to establish an upper bound on the RKHS norm based on additional assumptions, though. One thing that might help is that the eigenvalue decay of the Gaussian kernel is well understood (see Belkin, 2008).

References:

• Belkin, M. (2018). Approximation beats concentration? An approximation view on inference with smooth radial kernels. Proceedings of the 31st Conference On Learning Theory, in Proceedings of Machine Learning Research 75:1348-1361 Available from https://proceedings.mlr.press/v75/belkin18a.html.
• Steinwart, I., & Christmann, A. (2008). Kernels and Reproducing Kernel Hilbert Spaces. In Support Vector Machines (pp. 110–163). Springer. https://doi.org/10.1007/978-0-387-77242-4_4