Timeline for Non-analyticity of convolution
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Sep 3, 2012 at 7:33 | comment | added | flavio | Thanks a lot for your response. The idea of inversion was my first attempt. I do know the Fourier transform of $(x^2 - 1)^{1/2}x 1_{|x| > 1}$ but I don't get much further after that. Actually I'm looking at this in the framework of distribution theory where $g\in \mathcal{E}'(\mathbb{R})$. | |
| Sep 2, 2012 at 22:42 | comment | added | Zen Harper | Have you tried using a Taylor series expansion in $(z^2-1)^{1/2} = z(1-1/z^2)^{1/2}$, for $z = |x-y|>1$? Then, at least formally, you can get a decomposition into simpler operators, using a formal infinite sum. | |
| Sep 2, 2012 at 22:38 | comment | added | Zen Harper | I think it seems unlikely, without a lot of extra assumptions, that there will be any useful general statement. Knowing that g is not analytic at some point doesn't really give you much information, since there are many possible ways for analyticity to fail; and furthermore, one single value of g (maybe even all its derivatives) doesn't change f in any way (whereas by contrast, having g analytic at a point tells you something about g on a whole interval). Of course if you could invert the integral operator and express g directly in terms of f, you would get a lot of information. | |
| Aug 31, 2012 at 12:54 | answer | added | Alexandre Eremenko | timeline score: 3 | |
| Aug 31, 2012 at 11:55 | history | asked | flavio | CC BY-SA 3.0 |