Timeline for How to rigorously differentiate the convolution of a distribution and a $L^2$ function?
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 11, 2021 at 13:37 | vote | accept | Maximilian Janisch | ||
| Aug 10, 2021 at 18:07 | answer | added | Maximilian Janisch | timeline score: 1 | |
| Aug 10, 2021 at 18:05 | comment | added | Maximilian Janisch | @bathalf15320 Indeed I found what I needed (see my answer below). | |
| Aug 10, 2021 at 15:00 | comment | added | Maximilian Janisch | @bathalf15320 Thank you so much! I'll try to update you on if I found what I needed in the near future. | |
| Aug 9, 2021 at 15:00 | comment | added | bathalf15320 | jss100.campus.ciencias.ulisboa.pt where you will find the proceedings of a conference at Lisbon which include the article I referenced. I would recommend that you continue to publicações/textosdidacticos where you will find an accessible treatment to his approach to distributions, based on lectures he gave in Maryland. | |
| Aug 9, 2021 at 14:49 | comment | added | bathalf15320 | Sebastião e Silva's articles are available at the site | |
| Aug 9, 2021 at 14:48 | comment | added | bathalf15320 | Sebastão e silva's complete works are available at the site | |
| Aug 9, 2021 at 8:16 | comment | added | Maximilian Janisch | @bathalf15320 I was not able to find the article (all I found is this page where you can demand Stanford library to scan it for you, but it seems that is only for Stanford students). Do you have a hint for me? | |
| Aug 8, 2021 at 21:52 | comment | added | Maximilian Janisch | @bathalf15320 Thank you, I will look it up soon! | |
| Aug 8, 2021 at 21:51 | comment | added | Maximilian Janisch | @mathworker21 ahh I apologize for getting a bit pissed then 😅 | |
| Aug 8, 2021 at 4:57 | comment | added | bathalf15320 | If the convolution of distributions $f$ and $g$ is defined, then so is that of $Df$ (distibutional derivative) and $g$ and the expected formula holds. This can be found in the article "Integrals and orders of growth of distributions" by J. Sebastião e Silva which can be found online and contains an elementary (freshman analysis level) treatment of distributions, their definite (partial) integrals and convolutions. | |
| Aug 8, 2021 at 0:48 | comment | added | mathworker21 | @MaximilianJanisch nope, just an innocent question | |
| Aug 7, 2021 at 23:39 | comment | added | Maximilian Janisch | (And I meant to write $\langle T',\phi\rangle = - \langle T, \phi'\rangle$.) | |
| Aug 7, 2021 at 23:32 | comment | added | Maximilian Janisch | @mathworker21 (I do not know of a sensible way to define the convolution $T*f$ if $T$ is any tempered distribution and $f$ is any $L^2$-function.) | |
| Aug 7, 2021 at 23:31 | comment | added | Maximilian Janisch | @mathworker21 Your question seems a bit supercilious, but that might be just me. In any case, the Definition that I am using is the following: If $T\in C^\infty_{\text c}(\mathbb R;\mathbb C)^*$, then $T'$ is the distribution given by $$\langle T',\phi\rangle = - \langle T,\phi\rangle.$$ If you have some great insight or if there is an elementary fact that I am missing that would allow me to use this in order to conclude that $$(\mathscr K * f)' = \mathscr K' * f,$$ I would be happy to hear it. Note that $f$ is in $L^2$, so a special property of $\mathscr K$ is used. | |
| Aug 7, 2021 at 23:24 | comment | added | Maximilian Janisch | @ChristianRemling Thank you! I will check out the Coddington-Levinson book that you mentioned in the next days | |
| Aug 7, 2021 at 22:00 | comment | added | Christian Remling | You still have the usual (existence and uniqueness) theory available also for ODEs in this more general interpretation; this is discussed in many books, for example Coddington-Levinson. Your formula is then the variation-of-constants formula for the solutions of an inhomogeneous linear ODE. | |
| Aug 7, 2021 at 21:04 | comment | added | mathworker21 | do u know how $A'$ is defined? | |
| Aug 7, 2021 at 18:52 | history | asked | Maximilian Janisch | CC BY-SA 4.0 |