Timeline for Product of Heavisides: calculus vs Fourier transform vs wavefront set
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| May 23, 2022 at 0:46 | comment | added | Michael Engelhardt | If you use the word rigorous around here, you'd have to specify what space precisely you intend your $\Theta $ to be defined in. But for practical purposes, of course $\Theta^{2} =\Theta $, and the value at the one contentious point, $x=0$, should never matter for anything you calculate (you should be integrating against a sufficiently smooth function, so that the (finite) value at one point shouldn't matter). If you find your result does depend on $\Theta (x=0)$, then you've gone astray somewhere and you should investigate/regularize/consult your physics application as to the meaning. | |
| May 22, 2022 at 20:24 | history | edited | Evangeline A. K. McDowell | CC BY-SA 4.0 |
Clarify question
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| May 22, 2022 at 20:22 | comment | added | Evangeline A. K. McDowell | @MichaelEngelhardt the issue is that I am aware of three approaches, originally two methods that disagree + one method (wavefront) that does not work by assumption. Now only one method "works" (calculus one), but the other two don't. Actually, the "wrong" FT one I did matches "limiting sequences of smooth functions" (like $\frac{1}{2}(1+\tanh(x/n))$ which gives $\Theta(0) = 1/2$. My question is how to make sense of these differences. For example, should I believe or discard "calculus one" if I am being rigorous? | |
| May 22, 2022 at 14:07 | comment | added | Michael Engelhardt | I've lost track of what the question is. Originally, the question was that there were two mutually inconsistent proposals for what $\Theta^{2} $ means, and what the resolution of the inconsistency would be. Now, there is only one proposal. So, what is the question now? | |
| May 22, 2022 at 5:57 | comment | added | Carlo Beenakker | at $x=0$ your result $\Theta^2=\Theta/2$ does make sense (since $\Theta(0)=1/2$ is a natural definition) | |
| May 22, 2022 at 1:25 | history | edited | Evangeline A. K. McDowell | CC BY-SA 4.0 |
Fixing stupid mistake and updated the question
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| May 22, 2022 at 1:22 | comment | added | Evangeline A. K. McDowell | @WillieWong I was just testing this using Mathematica, and apparently it spits out something proportional to $\tilde{\Theta}$. You are right though that it's not locally integrable, it's my stupid miss. I will reframe the question. | |
| May 22, 2022 at 1:19 | comment | added | Gerry Myerson | The m.se post is math.stackexchange.com/questions/4413970/… | |
| May 22, 2022 at 1:08 | comment | added | Willie Wong | $i/k$ is not locally integrable. How did you compute the convolution of $\tilde{\Theta}$ against itself? | |
| S May 22, 2022 at 0:41 | review | First questions | |||
| May 22, 2022 at 3:10 | |||||
| S May 22, 2022 at 0:41 | history | asked | Evangeline A. K. McDowell | CC BY-SA 4.0 |