Timeline for Convolution definition in an old educational article
Current License: CC BY-SA 4.0
10 events
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May 17, 2022 at 2:14 | comment | added | ACR | I added a figure from the same paper for clarification. | |
May 17, 2022 at 2:13 | history | edited | ACR | CC BY-SA 4.0 |
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May 17, 2022 at 1:22 | comment | added | paul garrett | @LSpice, I agree, there are arguments in favor of this-and-that... depending on what ambient structure we want. But/and, then, for me, for clarity, I'd want to hear what the structural property is (your comment) first, and then see how to write a formula... that achieves it. E.g., for a group $G$ acting continuously on a TVS $V$, continuous compactly-supported functions naturally act by the integrated form. Ok. Then "convolution" of those functions $f,F$ should be such that $f\cdot (F\cdot v)=(f*F)\cdot v$ for all vectors $v$. From this we "discover" (two) formula(s) for convolution... :) | |
May 17, 2022 at 1:17 | comment | added | LSpice | On second glance (assuming, as @paulgarrett mentioned, that we are working on an Abelian group), the definition of $\varphi(\tau)$ is closer to that of the first $\theta(-\tau)$ than it is to the second $\theta(\tau)$. @paulgarrett, I think the difference is significant even for Abelian groups: consider, for example, the convolution of $\chi : t \mapsto e^{2\pi i t}$ with itself according to the $\varphi$ definition (giving $0$), and the usual definition (giving $\chi$). | |
May 17, 2022 at 0:00 | comment | added | ACR | @Lspice, these equations appear in separate sections under different headings with examples. | |
May 16, 2022 at 23:18 | comment | added | paul garrett | The first equation and second are not generally equivalent... Makes a person think there's a typo. But, unsurprisingly, on the real line or on circles (for periodic things) the group is abelian, so "inverse" is an automorphism, and it doesn't matter toooo much. | |
May 16, 2022 at 23:12 | history | edited | YCor |
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May 16, 2022 at 23:06 | comment | added | LSpice | How do you know it is not a typographical error? As you say, the context suggests that it is. | |
May 16, 2022 at 23:05 | history | edited | LSpice | CC BY-SA 4.0 |
Names of authors; link to article
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May 16, 2022 at 21:45 | history | asked | ACR | CC BY-SA 4.0 |