Timeline for Multivarate "RKHS" Examples
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 11, 2020 at 14:00 | comment | added | Christian Remling | The standard spaces of entire functions (de Branges space, or just Paley-Wiener space) give you $m=n=2$, and for higher values, you can just take orthogonal sums of these and interpret them as functions on $\mathbb C^N$. | |
| Sep 11, 2020 at 13:52 | comment | added | Nate Eldredge | Maybe I'm missing something, but why would it not work? Each coordinate function $f^{(i)}$ is $H^s(\mathbb{R}^n)$ and so it's continuous and its evaluation functions are bounded, and thus the evaluation functions for $f$ should likewise be bounded with a norm $\sqrt{m}$ times larger. | |
| Sep 11, 2020 at 13:44 | comment | added | ABIM | Oh really? I didn't know this worked when $m\neq 1$. Do you happen to have a reference? (I've only seen the recent paper of E. Novak ~ 2017) | |
| Sep 11, 2020 at 13:42 | comment | added | Nate Eldredge | What about Sobolev spaces $H^s(\mathbb{R}^n; \mathbb{R}^m)$ for $s > n/2$? I thought this was kind of the canonical example. | |
| Sep 11, 2020 at 12:57 | comment | added | ABIM | Unfortunately not, but I guess any such space should be of the form $H\cong H' \otimes \mathbb{R}^m$ for some (classical) RKHS H'. | |
| Sep 11, 2020 at 12:42 | comment | added | Matthew Daws | Do you have a reference for this sort of generalisation? | |
| Sep 11, 2020 at 12:30 | comment | added | ABIM | Oh for me its a hilbert space whose elements are functions (not to the base-field but to a topological vector space) and whose evaluation map is bounded linear operator | |
| Sep 11, 2020 at 12:07 | comment | added | Matthew Daws | What is your definition of a RKHS? Because wikipedia thinks a RKHS is a Hilbert space of real/complex-valued functions on a set. So I don't know how one could consider functions to $\mathbb R^m$; nor do I see how matrices fit into this framework, unless we think of matrices as functions from $[n^2]$ to $\mathbb C$? | |
| Sep 11, 2020 at 9:11 | history | asked | ABIM | CC BY-SA 4.0 |