Timeline for Prove $\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge {\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$
Current License: CC BY-SA 4.0
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| Aug 7, 2020 at 4:33 | history | edited | Tanya Vladi | CC BY-SA 4.0 |
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| S Aug 2, 2020 at 23:01 | history | bounty ended | CommunityBot | ||
| S Aug 2, 2020 at 23:01 | history | notice removed | CommunityBot | ||
| Jul 28, 2020 at 15:07 | comment | added | Tanya Vladi | @JohannesTrost, I understand, thank you for the clarification, however I am not interested in the tail, I need inequality to hold, you can think about it as for medium $\omega$, neither small nor large | |
| Jul 28, 2020 at 8:45 | comment | added | Johannes Trost | @TanyaVladi I was referring to your answer to user90189's comment.The constants for the tail $\sim \omega^{-\alpha-1}$ are given in the paper. | |
| Jul 28, 2020 at 2:13 | comment | added | Tanya Vladi | @JohannesTrost, thank you for the reference. I do not see how to use it though. It has a different representation, yes, very useful for asymptotics. I still do not see how I can get my lower bound. The truncation error cam provide some upper bound only. What am I missing? | |
| Jul 27, 2020 at 12:34 | history | edited | Rodrigo de Azevedo | CC BY-SA 4.0 |
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| Jul 27, 2020 at 11:41 | comment | added | Johannes Trost | ... in the paper arxiv.org/pdf/0911.4796.pdf that I recommended above, the asymptotics for your integral is $Q_{\beta}$, Eq. (13), The author's $\beta$ is your $\alpha$. | |
| Jul 27, 2020 at 10:35 | comment | added | Johannes Trost | Maybe this is helping for the large $\omega$ expansion: arxiv.org/pdf/0911.4796.pdf | |
| Jul 27, 2020 at 3:40 | history | edited | Tanya Vladi | CC BY-SA 4.0 |
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| S Jul 25, 2020 at 21:37 | history | bounty started | Tanya Vladi | ||
| S Jul 25, 2020 at 21:37 | history | notice added | Tanya Vladi | Draw attention | |
| Jul 23, 2020 at 14:46 | comment | added | Tanya Vladi | @user90189, thank you so much for the reference, all the bounds are are given without specific constants, here, I need that specific constant . I know as $i\to\infty$ $f(\omega) \sim \omega^{(-\alpha-1)}$, the next step is to get the constants for that tail. | |
| Jul 23, 2020 at 6:52 | comment | added | user90189 | More precise bounds are known; see e.g. Chen, Z-Q., Kim, P. and Song, R., Heat kernel estimates for the Dirichlet fractional Laplacian, 2010. | |
| Jul 22, 2020 at 22:43 | history | edited | Tanya Vladi | CC BY-SA 4.0 |
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| Jul 22, 2020 at 19:52 | history | edited | Tanya Vladi | CC BY-SA 4.0 |
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| Jul 22, 2020 at 19:45 | history | edited | Tanya Vladi | CC BY-SA 4.0 |
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| Jul 22, 2020 at 18:54 | history | edited | Rodrigo de Azevedo | CC BY-SA 4.0 |
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| Jul 22, 2020 at 17:44 | comment | added | Tanya Vladi | @CarloBeenakker, yes, thank you, it should be $\sqrt{\pi}$- I have edited the question | |
| Jul 22, 2020 at 17:43 | history | edited | Tanya Vladi | CC BY-SA 4.0 |
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| Jul 22, 2020 at 17:41 | comment | added | Carlo Beenakker | this inequality does hold ($\omega>0$, $1<\alpha<2$): $$\int_{0}^{\infty}\cos(\omega x)\exp(-x^{\alpha})dx \ge {\alpha^2 \sqrt{\pi} \over 8} \exp(-\omega^2/4) .$$ Equality is reached at $\alpha=2$. | |
| Jul 22, 2020 at 17:22 | comment | added | Carlo Beenakker | I presume your $dc$ should be $dx$; the inequality is badly violated; say for $\omega=1$ the l.h.s. and r.h.s. cross at $\alpha=1.4$. | |
| Jul 22, 2020 at 17:22 | comment | added | Michael Renardy | It is false for $\omega=0$, $\alpha=2$. | |
| Jul 22, 2020 at 17:07 | history | asked | Tanya Vladi | CC BY-SA 4.0 |