Timeline for Asymptotic decay rate of an oscillator integral
Current License: CC BY-SA 4.0
9 events
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| Jul 31 at 20:21 | comment | added | Willie Wong | @KoHey you need to be careful when comparing to the "continuous" case. The usual arguments for the $t^{(1-d)/2}$ decay for linear waves, via oscillatory integration, only works when one includes a smooth frequency cut-off. (For example, observe that the wave kernel is given by a distribution that cannot be represented as a bounded function at $t = 1$.) | |
| Jul 31 at 20:08 | answer | added | Willie Wong | timeline score: 4 | |
| Jul 31 at 17:46 | comment | added | Willie Wong | Your integral $I^d(t;v)$ is a continuous function in $v$ for any fixed $t$ and parameter $d$. The explicit values of $v = 0$ means that you cannot have a uniform in $v$ estimate of the form you claimed. The best you can do is to have the constant $C$ degenerating as $v \searrow 0$. On the other hand, if you impose a cut-off (replace $r^{d-2}$ by $r^{d-2} \phi(r)$ where $\phi:\mathbb{R}\to\mathbb{R}$ is smooth and compactly supported), then I think I can prove a uniform bound. | |
| Jul 31 at 17:04 | comment | added | Ko Hey | @WillieWong Thank you. The condition $v > 0$ may need. I think the integrand is similar to a solution for the continuous linear wave equation. The difference is only integral range. The claimed inequality holds in the continuous case. | |
| Jul 31 at 16:52 | comment | added | Willie Wong | The claimed inequality cannot hold. Consider the special case when $v = 0$. The $\theta$ integral yields a positive constant (the integrand is positive), so you are reduced to considering the integral $\int_0^{\sqrt{d}\pi} r^{d-2} \sin(tr) ~\mathrm{d}r$. This can be explicitly evaluated, and you find that asymptotic decay is no better than $1/t$. But I am wondering whether you actually want the expression you wrote down, or whether you want to impose a radial cut-off. | |
| Jul 31 at 16:40 | history | edited | Ko Hey | CC BY-SA 4.0 |
added 156 characters in body; edited tags
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| Jul 31 at 10:06 | comment | added | Ko Hey | @DavidLehavi I solve an initial value problem of discrete linear wave equation. The integral is given by the solution in polar coordinate. I want the decay estimate of it. | |
| Jul 31 at 10:01 | comment | added | David Lehavi | Could you give some context ? | |
| Jul 31 at 9:41 | history | asked | Ko Hey | CC BY-SA 4.0 |