Timeline for Is $\frac{\sin |\xi|}{|\xi|}$ in range of Fourier Transform for $n \ge 3$?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Nov 18, 2018 at 19:26 | vote | accept | sciona | ||
| Nov 18, 2018 at 19:24 | answer | added | Liviu Nicolaescu | timeline score: 10 | |
| Nov 18, 2018 at 17:39 | comment | added | sciona | @LiviuNicolaescu Gelfand and Shilov's book covers the $n = 2m+3$ case. How'd we argue for the even dimensions? | |
| Nov 18, 2018 at 17:06 | comment | added | Liviu Nicolaescu | Check Volume 1 of Gelfand and Shilov's book Generalized functions. You will find and explicit description of a distribution, concentrated along the unit sphere, whose Fourier transform is $(\sin|\xi|)/|\xi|$ | |
| Nov 18, 2018 at 17:00 | comment | added | Christian Remling | There is of course a (radial) distribution that has the desired FT, by Fourier inversion. The only meaningful question you can ask along these lines is how much regularity this distribution has. | |
| Nov 18, 2018 at 16:52 | comment | added | Yemon Choi | Just to expand very slightly on @YCor's comment/answer: because the Fourier transform is injective from $M({\bf R}^n)$ to $C_b({\bf R}^n)$, there is no $f\in L^1({\bf R}^3)$ with the properties that you desire, but instead you need to take the Fourier transform of a certain probability measure that is singular w.r.t. Lebesgue measure on ${\bf R}^3$. | |
| Nov 18, 2018 at 16:46 | history | edited | YCor |
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| Nov 18, 2018 at 16:46 | comment | added | YCor | For $n=3$, see mathoverflow.net/questions/315536 | |
| Nov 18, 2018 at 16:43 | history | edited | Liviu Nicolaescu | CC BY-SA 4.0 |
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| Nov 18, 2018 at 16:35 | review | First posts | |||
| Nov 18, 2018 at 17:23 | |||||
| Nov 18, 2018 at 16:31 | history | asked | sciona | CC BY-SA 4.0 |