Timeline for RKHS norm of Lipschitz functions
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 6, 2020 at 17:35 | vote | accept | Tyler6 | ||
| Jul 4, 2020 at 17:19 | comment | added | Nate Eldredge | I think another example is $\mathcal{H} = H^1_0((0,1])$, i.e. the absolutely continuous functions $f : [0,1] \to \mathbb{R}$ with $f' \in L^2([0,1])$, under the norm $\|f\|_{\mathcal{H}}^2 = \int_0^1 |f'|^2$. Then the "Lipschitz constant" is the Holder norm of exponent 1/2, $L = \sup |f(s)-f(t)|/\sqrt{|s-t|}$. So now let $f_n$ approach $f(x) = \sqrt{x}$ which has infinite $\mathcal{H}$-norm. | |
| Jul 4, 2020 at 13:37 | comment | added | DCM | What's your definition of a RKHS? I think the usual one is just "a Hilbert space of functions on a set whose evaluation functionals are norm continuous", no? | |
| Jul 4, 2020 at 13:06 | comment | added | Tyler6 | Hmm, correct me if I’m wrong but I thought $l^2$ wasn’t an RKHS? Either way it’s an interesting counterexample | |
| Jul 4, 2020 at 12:27 | history | answered | DCM | CC BY-SA 4.0 |