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Consider a Mercer Kernel $K\colon \mathcal{X}\times \mathcal{X}\to \mathbb{R}$, $\mathcal{X}$ being a compact subset of $\mathbb{R}^m$, and its (unique) associated Reproducing Kernel HIlbert Space $H_K$. Then $H_k\subset \mathcal{C}^0(\mathcal{X})$, where $\mathcal{C}^0(\mathcal{X})$ is the space of continuous function on $\mathcal{X}$, i.e., there exist some functions of $\mathcal{C}^0(\mathcal{X})$ which are not in $H_K$.

Could you give me an explanation of the latter statement? And/or could you give me an example of functions in $\mathcal{C}^0(\mathcal{X})$ which do not belong to $H_K$?

Thank you.

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  • $\begingroup$ What is the definition of a Mercer kernel? $\endgroup$
    – Andrei Kh
    Commented Mar 16, 2020 at 22:51

1 Answer 1

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What I know is that for some popular kernels such as SE kernels or Matern kernels, $H_k$ is dense in $C^{0}(\mathcal{X})$ with respect to sup-norm if $\mathcal{X}$ is compact (ref: Miccheli et al., 2006). This result can also be expanded to $L^p$-denseness (ref: Sriperumbudur et al., 2011). Thus it might be helpful for you to read these references.

Also, in Kanagawa et al., 2018, there is an explicit representation of RKHS $H_k$ using Fourier transform (Theorem 2.4) for stationary kernels. According to this reference (Remark 2.12), roughly speaking, a function $f$ is in $H_k$ if and only if $f \in C(\mathcal{X}) \cap L^2 (\mathcal{X})$ and $f$ is smooth enough (i.e., $\hat{f}(\omega)$ is small enough for large $\omega$) compared to the kernel $k$.

However, neither do I haven't seen explicit example of function $f$ that is not in RKHS generated by popular kernels such as SE kernels, but continuous.

I wish my answer would help you.

References:

Miccheli et al: https://www.jmlr.org/papers/volume7/micchelli06a/micchelli06a.pdf

Seriperumbudur et al: https://www.jmlr.org/papers/volume12/sriperumbudur11a/sriperumbudur11a.pdf

Kanagawa et al: https://arxiv.org/pdf/1807.02582

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