Timeline for Polynomial growth of Fourier transforms
Current License: CC BY-SA 3.0
12 events
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| Apr 16, 2013 at 21:01 | comment | added | Kevin Smith | Yes, this often happens when you are inverting Mellin transforms of step functions, but you're normally dealing with $x=\log y$, y>1. I have noticed you're edited answer below and I think it is a significant observation, but I am travelling at present and have not been able to correspond properly. I will ASAP. Thank you. | |
| Apr 16, 2013 at 18:45 | comment | added | Jens | Do you have an example of such an $F$ which is not in $L^1$? It seems like a fairly stringent condition (to ask that the partial integrals be bounded for each $x$). How do you know this is satisfied in the cases you are interested in? | |
| Apr 16, 2013 at 6:56 | comment | added | Kevin Smith | I assume that the limit exists for almost every $x$, and that it does not diverge for any $x$. | |
| Apr 15, 2013 at 17:36 | comment | added | Jens | I see. And are you assuming the limit $\lim_{R \to \infty} \int_{-R}^R F(y)e^{ixy}dy$ exists for every $x$, or just almost every $x$? | |
| Apr 15, 2013 at 10:54 | comment | added | Kevin Smith | Well, roughly speaking, a jump discontinuity at $x_0$ is given by $c_0H(x-x_0)$, where $H$ is Heaviside's step function, and a spike/$\delta$-function at $x_0$ is its derivative. Perhaps more formally you'd say that a jump discontinuity is defined as half the sum of the left and right limits. | |
| Apr 15, 2013 at 2:40 | comment | added | Jens | Could you clarify the distinction between jump discontinuities and 'spikes'? | |
| Apr 14, 2013 at 12:26 | history | edited | Kevin Smith | CC BY-SA 3.0 |
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| Apr 14, 2013 at 12:20 | history | edited | Kevin Smith | CC BY-SA 3.0 |
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| Apr 13, 2013 at 19:31 | answer | added | Bazin | timeline score: 4 | |
| Apr 13, 2013 at 16:14 | answer | added | Jens | timeline score: 3 | |
| Apr 13, 2013 at 15:23 | history | edited | Kevin Smith | CC BY-SA 3.0 |
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| Apr 13, 2013 at 15:15 | history | asked | Kevin Smith | CC BY-SA 3.0 |