Timeline for Properties of the Fourier Transform of Countably Supported Functions on $[0,1)$
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Dec 17, 2020 at 9:05 | comment | added | Mateusz Kwaśnicki | A formal remark: this should be denoted $\ell^p([0,1))$ rather than $\ell^p(\mathbb R/\mathbb Z)$, because the definition of the "Fourier transform" depends on this identification. | |
| Dec 16, 2020 at 22:35 | comment | added | MCS | Also, is there a simple characterization of the image of $\ell^{1}\left(\mathbb{R}/\mathbb{Z}\right)$ under this Fourier transform? | |
| Dec 16, 2020 at 22:33 | comment | added | MCS | Where is a specific citation for this particular "Wiener's Theorem"? (There are a lot of results named after him, and more than one of those is relevant to my current research)? As for I(i), I suppose the main issue here is to properly identify what, if any, Banach space we can use to make sense of $\hat{f}$ for square-summable. Also, on a technical point, isn't your "more conventional notation" different than what I've given: your object doesn't take finite values on its "support"; whereas mine does? | |
| Dec 16, 2020 at 22:23 | comment | added | Christian Remling | Question I(i) is unclear: obviously, the convergence won't be absolute if $f\notin\ell^1$, so order of summation matters, but there is no longer a distinguished natural ordering. | |
| Dec 16, 2020 at 22:20 | comment | added | Christian Remling | In more conventional notation, you are considering the FT of the measure $\sum f(t)\delta_t$. I(iii) holds (by a straightforward calculation); it is usually called Wiener's theorem. | |
| Dec 16, 2020 at 22:08 | history | asked | MCS | CC BY-SA 4.0 |