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Given that the convolution definition as far as I am aware is:

$(f*g)(t) = \int_{-\infty}^\infty f(\tau)g(t-\tau)d\tau$

Here I see that the functions f and g are being shifted across the time domain. I wonder if there is an equivalent for the scaling the time domain such at:

$(f?g)(t) = \int_{-\infty}^\infty f(\tau)g(t\tau)d\tau$

In my mind this would be similar to how a Fourier transform works (e.g. $g(t\tau)$ would be $cis(t\tau)$. I am not sure what kind of applications this could have in non-periodic data but I am still curious nontheless if there is a proper term for it.

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  • $\begingroup$ 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. $\endgroup$ Commented Sep 8, 2020 at 3:54
  • $\begingroup$ @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. $\endgroup$
    – Saxpy
    Commented Sep 8, 2020 at 6:31
  • $\begingroup$ 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. $\endgroup$
    – user131781
    Commented Sep 8, 2020 at 8:50
  • $\begingroup$ @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 $(\mathbb{R}^\times, \times)$). $\endgroup$ Commented Sep 8, 2020 at 14:28
  • $\begingroup$ @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. $\endgroup$
    – Saxpy
    Commented Sep 8, 2020 at 18:23

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