Timeline for Tanh version of a Fourier Transform?
Current License: CC BY-SA 3.0
17 events
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| S Nov 6, 2013 at 12:43 | history | suggested | Jean Van Schaftingen | CC BY-SA 3.0 |
corrected display of displayed math ($$)
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| Nov 6, 2013 at 12:22 | review | Suggested edits | |||
| S Nov 6, 2013 at 12:43 | |||||
| Jul 27, 2011 at 15:51 | history | edited | Bill Bradley | CC BY-SA 3.0 |
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| Jul 27, 2011 at 15:36 | comment | added | Bill Bradley | Dear pm, Thanks so much for the clarification of your exponential moment suggestion, I understand now. I hadn't explained #4 very clearly; I've taken another stab at it above. I am somewhat tempted to split this part off as a separate question. | |
| Jul 27, 2011 at 15:30 | history | edited | Bill Bradley | CC BY-SA 3.0 |
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| Jul 26, 2011 at 19:51 | comment | added | Marc Palm | Sorry, I did not directly see the question in your comment and my previous comment was also partly wrong. You are not getting moments, but exponential moments. First, $tanh(z) = 1-2 (1+ e^{2z})^{-1} = 1 + 2\sum\limits_{n>0} (-e^{2z})^n$ for $z<0$ and similarly for $x>0$. Then plugin $z=b+x$ and integrate summand for summand. So you get some Fourier expansion $\sum\limits_{n>0} a_n e^{bn}$ for $b<0$. Since $a_n$ is essentially the Laplace transform of $f$ at $n$, and I know at least that the question to recover $f$ from exponential moments should be discussed in the standard books. | |
| Jul 26, 2011 at 19:01 | comment | added | Marc Palm | So, I guess (2)-(3) are satisfactory answered, and for (1) you have the comment by Terry Tao. But for (4), I can only speak for myself, but I actually do not understand what you want here... Can you give the example for the construction of adding random variables with the Fourier transform explicitely? | |
| Jul 26, 2011 at 18:54 | comment | added | Bill Bradley | To Carnahan: In case the intention wasn't clear, I've restated the problem in (3) and (4). To pm: That approach sounds really interesting, but I'm getting confused on a few elementary points. How exactly are you getting the $2+2\sum_{n>0} EX^ne^{bn}$ expansion (since that doesn't look like the Taylor expansion of tanh). Also, given a linear combination of moments, is it in general possible to recover the original distribution uniquely? (And I guess in practice I would recover the distribution by sampling a finite set of points and using a semi-definite program?) Thanks, Bill | |
| Jul 26, 2011 at 18:25 | comment | added | Terry Tao | Indeed, $\overline{f}(a,b)$ is essentially the Radon transform of the two-variable function $f(x) \tanh(y)$, so the inverse Radon transform will do the trick. If one is only given $\overline{f}(1,b)$, this is essentially a convolution of f with the distribution $\tanh$, so a Fourier transform should be able to perform deconvolution as long as one avoids frequencies where the Fourier transform of $\tanh$ (or its derivative $\sech^2$, which is integrable) vanishes. | |
| Jul 26, 2011 at 17:54 | history | edited | Bill Bradley | CC BY-SA 3.0 |
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| Jul 26, 2011 at 17:05 | comment | added | Marc Palm | I guess $a$ and $b$ are real? Have you already exploited a series expansion $e^{\pm 2(ax\pm b)}$ for $\tanh$. Then this becomes essentially a question, which is equivalent to recover a probability distribution in terms of moment sums, from something like $2 +2 \sum_{n>0}EX^n e^{bn}$, which is holomorph in $b$ and where $EX^n$ is the $n$ th moment of your distribution. So in principle you can recover the moments $EX^n$, which is sufficient for the distribution of $X$, if it is distributed with total mass $1$. So $(3)$ yes! | |
| Jul 26, 2011 at 16:06 | comment | added | Julien Puydt | It doesn't look that ambitious to look for an inverse if you have an injective map and want to compute the antecedent of an element which is known to be in the image! | |
| Jul 26, 2011 at 8:31 | comment | added | S. Carnahan♦ | It seems rather ambitious to expect an inverse for a map from functions in one variable to functions in 2 variables. | |
| Jul 26, 2011 at 6:07 | comment | added | Andrew Homan | Yes. I don't have a reference for the inversion formula for the X-ray transform, but one should be able to follow the procedure for the Radon transform given here: wwwmath.uni-muenster.de/num/inst/natterer/Preprints/2000/… | |
| Jul 26, 2011 at 6:02 | comment | added | Andrew Homan | It looks like an attenuated (or weighted) X-ray transform. | |
| Jul 26, 2011 at 4:28 | comment | added | Helge | Is the integral understood as a principal value? If yes, can you prove that $\int| \overline{f}(a, b)|^2 db < \infty$ for some $a$ or $\int| \overline{f}(a, b)|^2 da < \infty$ for some $b$? | |
| Jul 26, 2011 at 3:33 | history | asked | Bill Bradley | CC BY-SA 3.0 |