I am working on some things that require Laplace, Fourier, and Mellin transforms (and a few others lurking in the background).

Simply put, if seems a transform is any function $G$ such that $F(g)=F(f(x))$, often in the form $F(g)=\int_{-\infty}^\infty f(x)q(x, g)dx$ or some such.

We could have simpler transforms: for any complex number $Z=u+iv$ we could have a transform $F(Z)\to(u+1)+i(v+1)$ It migh not have much use (maybe it does), but it is obvious it has certain useful qualities - e.g., it is bi-directional, unique, and has an inverse. On the other hand $F(Z)\to(u\cdot 0)+i(v+1)=i(v+1)$ "loses" a whole lot of information for no good purpose.

Is there a braoder theoretical basis for transforms - particularly but not only integral transforms of complex numbers - that drives their usefulness, or explains why 'transforms' are useful and well-behaved? Obviously, each example I gave has its mechanical benefits (turning convolution into multiplication, say), but I am seeking an understanding of the deeper abstract logic or structre that lets these things arise in the first place.

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    $\begingroup$ I think there is an "r" missing in the title of your question. Or maybe not $\ddot{\smile}$? $\endgroup$ – Rob Arthan May 13 '18 at 21:45
  • $\begingroup$ At its broadest, isn't this isomorphisms (or near-isomorphisms) of rings? (Without deviation into category theory, anyway.) $\endgroup$ – Chappers May 13 '18 at 21:45
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    $\begingroup$ This is quite a broad area. Specifically $q$ is usually referred to as a integration kernel: en.wikipedia.org/wiki/Integral_transform It sounds like you're interested in the algebraic properties of these, in which case you can read about Pontryagin duality, which explains how fourier transforms arise naturally from mapping abelian groups to their bidual: en.wikipedia.org/wiki/Pontryagin_duality $\endgroup$ – Alex R. May 13 '18 at 21:48
  • $\begingroup$ Sounds exactly like what I am looking for. Thanks|! $\endgroup$ – eSurfsnake May 13 '18 at 22:59
  • $\begingroup$ IMO, each transform has its own domain specific usefulness. However, I can make the general observation that both the Fourier Transform and Laplace Transform turn calculus operations (i.e. integration and differentiation) into algebraic operations. This makes them useful for solving differential equations in the transform domain. $\endgroup$ – Andy Walls May 14 '18 at 1:58

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