Timeline for Uncertainty principle (really for Mellin, but never mind that!)
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 12, 2012 at 10:14 | comment | added | H A Helfgott | I'm not sure how to get exponential decay on the boundary here (we rather get exponential decay on the bisectrix) or for that matter boundedness on the strip. Still, this sounds like a plausible direction... | |
| Nov 12, 2012 at 1:37 | comment | added | Ralph Furman | It's also known that if you have an entire function of a given exponential type, then if it is bounded on a wide enough sector then it has to be constant. | |
| Nov 12, 2012 at 1:36 | comment | added | Ralph Furman | There are some nice consequences of Phragmen-Lindelof (and its proof) along these lines in Titchmarsh's Theory of Functions. For example there's the following result of Carlson. Suppose $f(z)$ is holomorphic in some sector of interior angle $\theta$, is exponentially bounded $|f(z)|\ltlt e^{k|z|}$ and exponentially decays on the boundary. $\exp$ gives an example for $\theta<\pi$ and the result is that if $\theta=\pi$ then $f(z)\equiv0$. In fact, having such a non-trivial function would give you a stronger Phragmen-Lindelof principle that could prove that the exponential function does not e | |
| Nov 9, 2012 at 11:17 | comment | added | H A Helfgott | Also, I've been told there is a result by Hardy when the decay conditions are two-sided on both the $f(x)$ side and the $\widehat{f}(x)$, but it doesn't seem to answer the question here. | |
| Nov 9, 2012 at 11:15 | comment | added | H A Helfgott | I know Phragmen-Lindelof only as a tool to get subconvexity, so I am not familiar with the argument you allude to - could you give us a reference (or a one-line explanation)? | |
| Nov 9, 2012 at 2:21 | comment | added | H A Helfgott | Sorry, forgot to change variables. Meant $f(x)=e^{e^{−2x}}$. Agreed, it's not much better than (a), and the Fourier transform is slightly worse than (b). – H A Helfgott 0 secs ago | |
| Nov 9, 2012 at 0:57 | answer | added | Ralph Furman | timeline score: 3 | |
| Nov 8, 2012 at 19:55 | comment | added | Ralph Furman | How is $e^{-x^2}$ better than (a)? Also, $\hat f(t)$ decaying exponentially means that $f(x)$ extends to an analytic function on a horizontal strip and is $L^2$ on horizontal lines. Then the question comes down to finding a function analytic on a strip such that it decays very quickly on the positive real axis, and Phragmen Lindelof gives you bounds on how quickly it can decay (and how wide the strip may be) | |
| Nov 8, 2012 at 16:58 | comment | added | H A Helfgott | I can, by the way, do much better than (a) while doing not uch worse than (b): take $f(x) = e^{-x^2}$. | |
| Nov 8, 2012 at 15:51 | history | asked | H A Helfgott | CC BY-SA 3.0 |