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For a concise introduction to RKHS, you could have a look at sections 2.3 and 2.4 of Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences by Kanagawa et al. (2018).

In particular, they give a characterisation of the RKHS associated to a shift-invariant kernel on $\mathbb{R}^d$. In this case, the inner product is linked to the Fourier transform of the kernel:

Theorem 2Theorem 2.4 (Kanagawa et al.4)

Let $k(x, y) = \Phi(x - y)$ be a shift-invariant kernel on $\mathbb{R}^d$. Also and assume that $k \in C(\mathbb{R}^d) \cap L_1(\mathbb{R}^d)$.

Then the RKHS of k is given by

$$ \mathcal{H}_k = \Big\lbrace f \in C(\mathbb{R}^d) \cap L_1(\mathbb{R}^d), ~||f||_{\mathcal{H}_k} < \infty \Big\rbrace $$

$$ \mathcal{H}_k = \Big\lbrace f \in C(\mathbb{R}^d) \cap L_1(\mathbb{R}^d), ~||f||_{\mathcal{H}_k} < \infty \Big\rbrace $$ Where Where the inner product is given by $$ \langle f, g \rangle_{\mathcal{H}_k} = \frac{1}{(2\pi)^{d/2}}\int\frac{\mathcal{F}[f](\omega)\overline{\mathcal{F}[g](\omega)}}{\mathcal{F}[\Phi](\omega)}d\omega $$

$$ \langle f, g \rangle_{\mathcal{H}_k} = \frac{1}{(2\pi)^{d/2}}\int\frac{\mathcal{F}[f](\omega)\overline{\mathcal{F}[g](\omega)}}{\mathcal{F}[\Phi](\omega)}d\omega $$


If you're working with more exotic spaces or kernels (complex or bounded spaces, periodic kernels, ...) you could have a look at the book by Berlinet and Thomas-Agnan Reproducing Kernel Hilbert Spaces in Probability and Statistics.

Chapter 7.4 gives a list of RKHSs together with associated kernel and analytical expression for the inner product.

Finally, Chapter 6.2 mentions a numerical scheme for computing the RKHS norm in a special case. Maybe this could be interessting to you.

For a concise introduction to RKHS, you could have a look at sections 2.3 and 2.4 of Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences by Kanagawa et al. (2018).

In particular, they give a characterisation of the RKHS associated to a shift-invariant kernel on $\mathbb{R}^d$. In this case, the inner product is linked to the Fourier transform of the kernel:

Theorem 2.4

Let $k(x, y) = \Phi(x - y)$ be a shift-invariant kernel on $\mathbb{R}^d$. Also assume that $k \in C(\mathbb{R}^d) \cap L_1(\mathbb{R}^d)$.

Then the RKHS of k is given by

$$ \mathcal{H}_k = \Big\lbrace f \in C(\mathbb{R}^d) \cap L_1(\mathbb{R}^d), ~||f||_{\mathcal{H}_k} < \infty \Big\rbrace $$ Where the inner product is given by $$ \langle f, g \rangle_{\mathcal{H}_k} = \frac{1}{(2\pi)^{d/2}}\int\frac{\mathcal{F}[f](\omega)\overline{\mathcal{F}[g](\omega)}}{\mathcal{F}[\Phi](\omega)}d\omega $$

If you're working with more exotic spaces or kernels (complex or bounded spaces, periodic kernels, ...) you could have a look at the book by Berlinet and Thomas-Agnan Reproducing Kernel Hilbert Spaces in Probability and Statistics.

Chapter 7.4 gives a list of RKHSs together with associated kernel and analytical expression for the inner product.

Finally, Chapter 6.2 mentions a numerical scheme for computing the RKHS norm in a special case. Maybe this could be interessting to you.

For a concise introduction to RKHS, you could have a look at sections 2.3 and 2.4 of Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences by Kanagawa et al. (2018).

In particular, they give a characterisation of the RKHS associated to a shift-invariant kernel on $\mathbb{R}^d$. In this case, the inner product is linked to the Fourier transform of the kernel:

Theorem 2.4 (Kanagawa et al.)

Let $k(x, y) = \Phi(x - y)$ be a shift-invariant kernel on $\mathbb{R}^d$ and assume $k \in C(\mathbb{R}^d) \cap L_1(\mathbb{R}^d)$.

Then the RKHS of k is given by

$$ \mathcal{H}_k = \Big\lbrace f \in C(\mathbb{R}^d) \cap L_1(\mathbb{R}^d), ~||f||_{\mathcal{H}_k} < \infty \Big\rbrace $$

Where the inner product is given by

$$ \langle f, g \rangle_{\mathcal{H}_k} = \frac{1}{(2\pi)^{d/2}}\int\frac{\mathcal{F}[f](\omega)\overline{\mathcal{F}[g](\omega)}}{\mathcal{F}[\Phi](\omega)}d\omega $$


If you're working with more exotic spaces or kernels (complex or bounded spaces, periodic kernels, ...) you could have a look at the book by Berlinet and Thomas-Agnan Reproducing Kernel Hilbert Spaces in Probability and Statistics.

Chapter 7.4 gives a list of RKHSs together with associated kernel and analytical expression for the inner product.

Finally, Chapter 6.2 mentions a numerical scheme for computing the RKHS norm in a special case. Maybe this could be interessting to you.

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For a concise introduction to RKHS, you could have a look at sections 2.3 and 2.4 of Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences by Kanagawa et al. (2018).

In particular, they give a characterisation of the RKHS associated to a shift-invariant kernel on $\mathbb{R}^d$. In this case, the inner product is linked to the Fourier transform of the kernel:

Theorem 2.4

Let $k(x, y) = \Phi(x - y)$ be a shift-invariant kernel on $\mathbb{R}^d$. Also assume that $k \in C(\mathbb{R}^d) \cap L_1(\mathbb{R}^d)$.

Then the RKHS of k is given by

$$ \mathcal{H}_k = \Big\lbrace f \in C(\mathbb{R}^d) \cap L_1(\mathbb{R}^d), ~||f||_{\mathcal{H}_k} < \infty \Big\rbrace $$ Where the inner product is given by $$ \langle f, g \rangle_{\mathcal{H}_k} = \frac{1}{(2\pi)^{d/2}}\int\frac{\mathcal{F}[f](\omega)\overline{\mathcal{F}[g](\omega)}}{\mathcal{F}[\Phi](\omega)}d\omega $$

If you're working with more exotic spaces or kernels (complex or bounded spaces, periodic kernels, ...) you could have a look at the book by Berlinet and Thomas-Agnan Reproducing Kernel Hilbert Spaces in Probability and Statistics.

Chapter 7.4 gives a list of RKHSs together with associated kernel and analytical expression for the inner product.

Finally, Chapter 6.2 mentions a numerical scheme for computing the RKHS norm in a special case. Maybe this could be interessting to you.