# The RKHS with kernel $\exp\{x y\}$

I would like to understand the reproducing kernel Hilbert space (RKHS) $$\mathcal{H}$$ of real valued functions defined on $$\mathbb{R}$$ generated by or associated with the kernel $$K(x,y) = \exp\{x y\} \text{ for }x,y\in\mathbb{R}.$$

This is a "real" version of Segal-Bargmann space but I am not sure what properties carry over from the complex setting.

## Questions

• What are the functions contained in $$\mathcal{H}$$? Can they be characterised by an integral conditions such as for Segal-Bargmann?
• Is there an orthonormal Eigenbasis from Mercer's theorem, even though the domain of $$K$$ is non-compact and the kernel not bounded?
• Is there an explicit form for the Eigenbasis?
• You should use $( \ \ )$ because $\{ \ \ \}$ is unused and might mean the fractional part Aug 14, 2019 at 13:52
• How about exp(-xy)? Is that a valid kernel?
– anmo
Apr 29, 2021 at 14:46
• No, I fear it is not!
– g g
Apr 30, 2021 at 7:39

The structure of $$\mathscr{H}$$ is closely related to the structure of the RKHS $$\mathscr{G}$$ generated by the Gaussian kernel and $$\mathscr{G}$$ has been thoroughly analysed.

## Relation to the Gaussian kernel

It is straightforward to verify that the kernels $$K(x,y)=\exp\left\{x y\right\}$$ of $$\mathscr{H}$$ and $$G(x,y)=\exp\left\{-\frac{1}{2}(x - y)^2\right\}$$ of $$\mathscr{G}$$ are related by: $$K(x,y) =f(x) G(x,y) f(y) \text{ with } f(x) = \exp\left\{\frac{1}{2}x^2\right\}.$$

## Relation between $$\mathscr{H}$$ and $$\mathscr{G}$$

From the relation above between the kernels follow the following facts about the spaces and the scalar product (see e.g. Paulsen Proposition 5.20):

1. $$\mathscr{H}=\left\{fg\mid g \in \mathscr{G}\right\}$$
2. $$_\mathscr{H} \,=\, _\mathscr{G}.$$

## Facts about $$\mathscr{G}$$ and conclusions

In Steinwart-Christmann Theorem 4.42 $$\mathscr{G}$$ is characterised by:

1. The functions $$\tilde{e}_n=\frac{1}{\sqrt{n!}}x^n\exp\left\{-\frac{1}{2}x^2\right\}$$ are an ONB of $$\mathscr{G}.$$
2. $$\mathscr{G}=\left\{g=\sum{\alpha_n \tilde{e}_n\mid \sum \alpha_n^2 < \infty}\right\}$$

and one can conclude about $$\mathscr{H}:$$

1. The functions $$e_n=\frac{1}{\sqrt{n!}}x^n$$ are an ONB of $$\mathscr{H}.$$
2. $$\mathscr{H}=\left\{h=\sum{\frac{\alpha_n }{\sqrt{n!}}x^n\mid \sum \alpha_n^2 < \infty}\right\}$$.

The series can be compared to the Taylor expansion of $$h\in\mathscr{H}$$ to conclude that $$\mathscr{H}$$ consists of real analytic functions such that $$\sum{\frac{(h^{(n)}(0))^2}{n!} }<\infty.$$