Timeline for What is the term for convoluting but scaling the time domain instead of shifting?
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| Sep 8, 2020 at 18:23 | comment | added | Saxpy | @Mark I see. I need to do a bit more studying as I am not familiar with the term locally compact abelian group. That being said, Mellin convolution was in fact the term I was looking for, thank you. I will look into this topic further. | |
| Sep 8, 2020 at 14:28 | comment | added | Mark Schultz-Wu | @Saxpy The Mellin transform, like the Fourier transform, is defined for a single function. Mellin convolution, like Fourier convolution, is defined for pairs of functions. The particular convolution is $(f\ast_{\mathrm{Mellin}}g)(y) = \int_{0}^\infty f(x) g(y/x) \frac{\mathrm{d} x}{x}$, which can be found on the page I linked (or here). As mentioned, you can preform this analogous setup over other locally compact abelian groups (other than $(\mathbb{R}, +)$ or $(\mathbb{R}^\times, \times)$). | |
| Sep 8, 2020 at 8:50 | comment | added | user131781 | The convolution can be defined for functions on groups, typically abelian locally compact groups with Haar measure. The cases you mention are those of the real line, with addition, and the positive reals with multiplication. The general situation is covered in any text on abstract harmonic analysis, as mentioned above. | |
| Sep 8, 2020 at 6:31 | comment | added | Saxpy | @Mark Is there something analogous for two separate functions? After looking at the Mellin transform it looks like this is for a single function much like a fourier transform or laplace transform. | |
| Sep 8, 2020 at 3:54 | comment | added | Mark Schultz-Wu | The "multiplicative" version of the Fourier transform is the Mellin transform. This has a corresponding convolution theorem (ctrl+F "mellin convolution"), which obeys $\mathcal{M}[f]\mathcal{M}[g] = \mathcal{M}[f\ast_{\mathrm{Mellin}}g]$. In generally one can define an analogue of the Fourier transform (which leads to a convolution theorem) for examining symmetries other than "Shifting" or "Scaling". This is done in Harmonic Analysis. | |
| Sep 8, 2020 at 2:40 | review | First posts | |||
| Sep 8, 2020 at 5:54 | |||||
| Sep 8, 2020 at 2:37 | history | asked | Saxpy | CC BY-SA 4.0 |