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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?
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  • $\begingroup$ You should use $( \ \ )$ because $\{ \ \ \}$ is unused and might mean the fractional part $\endgroup$
    – reuns
    Aug 14, 2019 at 13:52
  • $\begingroup$ How about exp(-xy)? Is that a valid kernel? $\endgroup$
    – anmo
    Apr 29, 2021 at 14:46
  • $\begingroup$ No, I fear it is not! $\endgroup$
    – g g
    Apr 30, 2021 at 7:39

1 Answer 1

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Selfanswer!

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. $<fg_1, fg_2>_\mathscr{H} \,=\, <g_1,g_2>_\mathscr{G}.$

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 $$ \sum{\frac{(h^{(n)}(0))^2}{n!} }<\infty.$$

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