Timeline for Asymptotics of an entire function with real zeroes on the real line
Current License: CC BY-SA 4.0
15 events
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| Sep 29 at 19:36 | comment | added | Synia | Good question. I am afraid I still don't know yet what will be the critical properties, but you can have an idea of the zeroes by looking at "determinantal point processes" (see e.g. en.wikipedia.org/wiki/Determinantal_point_process or arxiv.org/abs/0911.1153). I was more hoping to have from the start a list of desirable properties that I could check by hand. But maybe this can also work the other way round. In fact, supposing that one has $ \alpha_k = k^\gamma + X_k $ (symmetrised for $ k < 0 $) where $ (X_k)_k $ is an i.i.d. sequence of Gaussians (say) would be a good start. | |
| Sep 28 at 12:35 | comment | added | Alexandre Eremenko | @Synia: the main question is not about conclusions but about assumptions: what kind of property of zeros are you willing to use? Asymptotics of zeros is clearly not enough to make such conclusions. | |
| Sep 27 at 13:18 | comment | added | Synia | Last question then: do you think there should be one/did you look for one such quantity at some point? Maybe an inequality on $ (M, +\infty) $ for a given $ M $ that one can compute explicitly that states that $ \ln|f(x)| \leq C \exp(-h_1 x^\alpha) $ here again with an explicit $C$? | |
| Sep 26 at 16:29 | comment | added | Alexandre Eremenko | @Synia, No, I don't think there is a ready reference for a bound for the norm in terms of zeros. | |
| Sep 26 at 16:28 | history | edited | Alexandre Eremenko | CC BY-SA 4.0 |
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| S Sep 26 at 13:53 | history | edited | gmvh | CC BY-SA 4.0 |
I just corrected the name of Lindelöf and added a bar to \theta
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| S Sep 26 at 13:53 | history | suggested | Synia | CC BY-SA 4.0 |
I just corrected the name of Lindelöf and added a bar to \theta
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| Sep 26 at 13:05 | review | Suggested edits | |||
| S Sep 26 at 13:53 | |||||
| Sep 26 at 13:04 | comment | added | Synia | Indeed, but afterwards, I need to estimate the $ L^p $ norm in a certain way to prove that it has some property (the roots are random, and the $ L^p $ norm of the random function should be a random variable with some moments), so maybe the expression will help in some sense. In fact, this is another question of interest to me: is there a bound on the $L^p$ norm using some good quantity involving the zeroes. Is it somewhere in the book of Levin ? | |
| Sep 25 at 16:59 | history | edited | Alexandre Eremenko | CC BY-SA 4.0 |
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| Sep 25 at 16:51 | comment | added | Alexandre Eremenko | Exact expressions for $h_1$ and $h_2$ are not important for your question about $L^p$. What is important is that they are negative, and this is geometrically evident, provided that both $a$ and $b$ are positive. In case $ab=0$ you have Airy like behavior of $\log|f|$ and additional information on asymptotics of zeros is needed. | |
| Sep 25 at 13:06 | vote | accept | Synia | ||
| Sep 25 at 12:54 | comment | added | Synia | Thanks a lot!!! I was looking at the book by Boas, but I had planned to look at Levin at some point. I also understand why, in the case of the Airy function, there is no exponential for $x\to-\infty$ (this is the case where $a=0$). Let me look at the chapter II in the book of Levin, and I will see if I manage to compute myself the $h_i$'s. And yes, this is really what I need (for some random zeroes...). | |
| Sep 25 at 12:32 | history | edited | Alexandre Eremenko | CC BY-SA 4.0 |
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| Sep 25 at 12:17 | history | answered | Alexandre Eremenko | CC BY-SA 4.0 |