Timeline for Fourier transform of (real) exponential
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| S May 22, 2020 at 10:38 | history | suggested | user142929 | CC BY-SA 4.0 |
Added a capital letter in the title.
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| May 22, 2020 at 8:28 | review | Suggested edits | |||
| S May 22, 2020 at 10:38 | |||||
| May 22, 2020 at 8:06 | answer | added | Salviati | timeline score: 0 | |
| Mar 12, 2013 at 11:46 | answer | added | user80744 | timeline score: 0 | |
| Aug 7, 2011 at 3:50 | comment | added | Phil Isett | There's no need to introduce hyperfunctions -- Nate Eldredge's answer is already correct. A smooth test function of compact support has a Fourier transform which is naturally an entire function (defined by the same formula as the usual Fourier transform). Integrating against, say, $e^x$ will just give you the value at $-i$ as you can see from the formula (depending on your normalizations), just as you would expect. It already makes sense -- in my opinion, you don't honestly need to know the "correct" space of test functions until you run into a specific problem. | |
| Jun 3, 2011 at 13:36 | answer | added | Michael Renardy | timeline score: 2 | |
| Jun 3, 2011 at 12:59 | answer | added | paul garrett | timeline score: 4 | |
| Oct 14, 2010 at 23:44 | history | edited | Willie Wong |
edited tags
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| Oct 13, 2010 at 13:50 | answer | added | Richard Borcherds | timeline score: 13 | |
| Oct 13, 2010 at 11:49 | comment | added | Tim van Beek | Hyperfunctions are an interesting hint for this question, but as far as I know the Fourier transform is defined for hyperfunctions of sub-exponential growth only (see Kaneko, Arscott: "Introduction int o hyperfunctions"). (A functional like $\delta_i$ is not a hyperfunction.) | |
| Oct 13, 2010 at 3:32 | comment | added | Nate Eldredge | Since $e^{iax}$ has Fourier transform $\delta_a$, we formally might think of taking $a = -i$, suggesting that the Fourier transform of $e^x$ "should be" $\delta_{-i}$, i.e. a functional taking $f$ to $f(-i)$. To me, this suggests that the right domain of test functions for such a functional should be holomorphic functions. It looks like the theory of hyperfunctions (en.wikipedia.org/wiki/Hyperfunction) treats this; I don't know anything about them, but it may be worth a look. | |
| Oct 13, 2010 at 2:54 | answer | added | Helge | timeline score: -6 | |
| Oct 13, 2010 at 1:30 | history | asked | johny | CC BY-SA 2.5 |