Timeline for Fréchet derivative of evaluation-like functional (multivariate)
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Sep 11, 2020 at 19:31 | comment | added | Daniele Tampieri | @AnnieLeKatsu I suspect it is not so: as a matter of fact $$ (f+\varepsilon g)\circ (f +\varepsilon g)\neq f\circ f+\varepsilon g\circ g.$$ However, despite the presumably higher complexity of calculations, the same method for the calculation of functional derivative is applicable. Just the computations are more tedious. | |
| Sep 11, 2020 at 13:38 | comment | added | ABIM | How much would it change if $n=m$ and we evalue $f\circ f$ instead of $f$ at x? We only switch $J_f$ for $J_{f\circ f}?$ | |
| Aug 25, 2020 at 4:32 | comment | added | Daniele Tampieri | @AnnieLeKatsu, better "Weighted Hilbert space". In general Hörmander spaces are (weighted) Hilbert spaces only if their associated Lebesgue space is $L^2$: otherwise they are only Banach. By googling "Weighted Hilbert space" you should be able to choose from a wide range of constructions and find what you need. | |
| Aug 24, 2020 at 20:33 | comment | added | ABIM | Ah this is nice, you mean googling Hörmander spaces? I've only ofund some russian texts... | |
| Aug 24, 2020 at 20:18 | comment | added | Daniele Tampieri | @AnnieLeKatsu, after thinking a bit, I noticed that the structure of Hilbert space you provided on $C^1(\Bbb R^n,\Bbb R^m)$ is quite common: indeed it is a weighted Hilbert space with weight $\|x\|$. This kind of structure was used also by Hörmander for the solution of the $\bar\partial$-equation, therefore is quite "popular": by googling that name, you'll probably find a lot of information on spaces closer to the one you use than Segal-Bargmann ones I associated by analogy. | |
| Aug 24, 2020 at 12:01 | comment | added | Daniele Tampieri | @AnnieLeKatsu the answer is still no: however, the inner product you provided reminds me of (as it is very similar to) the one defined on the Segal-Bargmann space, which is a Hilbert space of entire functions and has recently been investigated extensively by Brian C. Hall. | |
| Aug 24, 2020 at 11:12 | comment | added | ABIM | Oh I meant $C^1$ with topology induced by the inner product I provided. | |
| Aug 24, 2020 at 10:43 | comment | added | Daniele Tampieri | Thank you, @AnnieLeKatsu. Regarding the linear product space $C^1(\Bbb R^n, \Bbb R^m)\times\Bbb R^n$ I am not aware of other uses and I haven’t seen it used in the literature. It’s a very particular space since it is the product between a infinite dimensional Hilbert space and a finite dimensional one, and I constructed (or better guessed)the structure of its topology from the one you gave to $C^1$ and the standard $\Bbb R^n$ one. | |
| Aug 24, 2020 at 10:14 | comment | added | ABIM | Great answer Daniele. May I ask, have you ever come across this space/ is it a well-studied object? | |
| Aug 22, 2020 at 12:47 | vote | accept | ABIM | ||
| Aug 21, 2020 at 19:17 | history | answered | Daniele Tampieri | CC BY-SA 4.0 |