Timeline for Integration for Dirac-delta function
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Mar 18, 2019 at 21:43 | comment | added | Liviu Nicolaescu | @user131781 The composition with a diffeomorphism was know by Laurent Schwarz since the 40s when he created the theory of distributions and it is a rather trivial fact. (Change in variables formula) If your G were a diffeomorphism you would not have asked the question on this site because you already knew the answer. What happens when G is not a submersion is more interesting and that is where the reference to Hormander becomes relevant. | |
| Mar 18, 2019 at 9:38 | comment | added | user131781 | @Liviu Nicolaescu Reading Hörmander is always a pleasure and I have spent many hours doing so, even without being commanded to by you. However, since I pointed to a rigorous definition which has been around since the time when he was still a young student I don‘t see the relevance to my answer. Also I rather think that using his monograph for an elementary question in one variable calculus is a bit of overkill. | |
| Mar 17, 2019 at 21:48 | comment | added | Liviu Nicolaescu | @user131781 Read Chapter VI in volume 1 of Hormander's 4 volume opus on partial differential equations. One can define the pullback of any smooth function by any smooth map. For distributions you cannot do this in genera.l Hormander explains when this is possible in terms of wave front sets. This condition is trivial;ly satisfied when you pullback by submersions. Beyond submersions you need to check Hormander's condition on wave front sets. | |
| Mar 8, 2019 at 7:53 | vote | accept | user136540 | ||
| Mar 7, 2019 at 11:17 | comment | added | Liviu Nicolaescu | @user131781. You gotta make up your mind. You said below that $G$b is not a nice function? What do you mean by that. Without any specific assumptions tthe best answer you can expect goes along tthe lines of another less known Heisenberg principle "You can say everything about nothing and nothing about everything." This is a math site. A bit of rigor and precision is expected of you. | |
| Mar 7, 2019 at 10:52 | comment | added | user131781 | @LiviuNicolaescu This implies that the formulae given here are perfectly valid provided the usual proviso „under suitable conditions on the functions“ is added. These can be made precise (and are less severe than some mentioned here) but I suspect that is not a major preoccupation of the OP. In order to use your method, one requires not just a rigorous definition but a result of the type: when $f$ converges in some distributional sense, then so does £f\circ g$. Could you give a reference and precise statement for the one you are using? | |
| Mar 7, 2019 at 10:34 | comment | added | user131781 | @LiviuNicolescu | |
| Mar 7, 2019 at 10:33 | comment | added | user131781 | @LiviuNicolescu The pullback of a distribution is not a rather ill defined object—-it was defined and studied in some detail in the 50‘s for the case where $g$ is a diffeomorphism by, amongst others, Heinz König and J. Sebastião e Silva. Using a standard localisation process („recollement des morceaux“), it can be extend to much more general situations. Thus $\delta \circ \sin (x)$ is well-defined since we can find a covering of the real line by open intervals on each of which either the sine function is a diffeomorphism or it takes its values in an open interval on which $\delta$ is smooth. | |
| Mar 5, 2019 at 2:26 | comment | added | user136540 | @LiviuNicolaescu To calculate the above equation in a numerical way, I also tried to replace $\delta$ as a Gaussian, but the magnitude of tiny variation have to be decided according to the number of samples for numerical integration. Could you tell me your suggestion to decide the tiny variation? any rule-of-thumb. | |
| Mar 4, 2019 at 14:39 | comment | added | Liviu Nicolaescu | @CarloBeenaker You have to know that I, as a pure mathematician, am a bit envious on theoretical physicists that integrate objects like $\delta(g(x))$ to get spectacular conclusions. To me $\delta(g(x))$ is the pullback of a distribution which is a rather ill defined operation, unless $0$ is a regular value of $g$. Suddenly I get worried. For my sanity I replace $\delta$ by a Gaussian with tiny variation and my worries disappear. | |
| Mar 4, 2019 at 11:17 | comment | added | lcv | Carlo Beenakker, @LiviuNicolaescu I believe the answer is simpler than that :). | |
| Mar 4, 2019 at 11:01 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
added 76 characters in body
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| Mar 4, 2019 at 11:00 | comment | added | Liviu Nicolaescu | @user136540 You have to be ca Dirac function is not a function, it is a generalized function. The next best thing you can do is approximate $\delta$ by a Gaussian with a a tiny, tiny variance. | |
| Mar 4, 2019 at 9:59 | comment | added | user136540 | I deeply appreciate your response. If I want to calculate the integration by the numerical way such as Monte Carlo simulation. I mean random sampling on PDF of x. What should I do? In addition, Could you explain the meaning of the equation you wrote? I cannot fully understand it. | |
| Mar 4, 2019 at 9:17 | history | answered | Carlo Beenakker | CC BY-SA 4.0 |