Timeline for Examples of RKHS that are "classical"
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Aug 8, 2023 at 6:13 | comment | added | lost_analyst | @YemonChoi + YonahBorns-Weil: Thank you for the suggestions! I guess what I was looking for is a rigorous statement of these claims. In fact, I finally found some details here: math.uh.edu/~vern/rkhs.pdf If either of you would like to your own details as an answer, I will happily accept it. Otherwise, I might self-answer with this link. | |
| Aug 8, 2023 at 1:46 | comment | added | Yemon Choi | What I'm trying to say is: if your original inner product space of functions is indeed a Hilbert space, the chances are that this can only be proved when point evaluations are continuous, i.e. if you have a RKHS. Now as I said above this is not a proof, but it's an attempt to give some justfiication for @YonahBorns-Weil's comment (which I tend to agree with). | |
| Aug 8, 2023 at 1:45 | comment | added | Yemon Choi | A non-rigorous POV that I think is still useful: if you have some space of functions on a set $X$, how do you know if it is complete w.r.t. a given inner product? You'll need to show that given a Cauchy sequence $(f_n)$ there is some genuine function $f$ on $X$ that is the norm limit of the $f_n$, and in many cases the only way to define $f(x)$ is to know that the sequence $f_n(x)$ is Cauchy. But how do you get pointwise Cauchy estimates from a Cauchy norm condition? Well, you probably want "evaluation at x" to be continuous w.r.t. your norm -- and that is the RKHS condition. | |
| Aug 7, 2023 at 9:34 | comment | added | Igor Khavkine | For the uninitiated, presumably RKHS stands for Reproducing Kernel Hilbert Space? | |
| Aug 7, 2023 at 4:41 | comment | added | Nate Eldredge | Morrey's inequality tells you which Sobolev spaces embed into the space of continuous functions, and thus are RKHSs. $L^2(X,\mu)$ is an RKHS if and only if $(X,\mu)$ is purely atomic. | |
| Aug 7, 2023 at 4:16 | comment | added | Yonah Borns-Weil | Honestly pretty much any Hilbert space of functions you'll come across is a RKHS (Lest we forget, $L^2$ spaces are not actually spaces of functions). | |
| S Aug 7, 2023 at 3:54 | review | First questions | |||
| Aug 7, 2023 at 5:04 | |||||
| S Aug 7, 2023 at 3:54 | history | asked | lost_analyst | CC BY-SA 4.0 |