Timeline for Functions belong to $L^{\frac{2n}{n+1}}$ whose Fourier transforms are infinite on $S^{n-1}$
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 11, 2018 at 1:01 | vote | accept | Tomas | ||
| Sep 10, 2018 at 20:47 | comment | added | Mateusz Kwaśnicki | I suppose this example can be made even more convincing by showing that the distributional Fourier transform of $F(|x|)$, or the distributional Hankel transform of $F(r)$, coincides with a continuous function (in the extended sense, that is, with values in $[-\infty, \infty]$). The usual integration by parts trick (a.k.a. Abel–Dirichlet test) should work here: $r^{n/2} (2+r)^{-n/2} / \log(2+r)$ eventually decreases to zero, while the indefinite integral of $J_{n/2-1}(r) J_{n/2-1}(r s)$ is bounded unless $s = 1$. But I did not attempt to work out the details. | |
| Sep 10, 2018 at 17:17 | history | edited | Terry Tao | CC BY-SA 4.0 |
added 58 characters in body
|
| Sep 10, 2018 at 17:12 | history | answered | Terry Tao | CC BY-SA 4.0 |