# Laplace, Legendre, Fourier, Hankel, Mellin, Hilbert, Borel, Z…: unified treatment of transforms?

I understand “transform methods” as recipes, but beyond this they are a big mystery to me.

There are two aspects of them I find bewildering.

One is the sheer number of them. Is there a unified framework that includes all these transforms as special cases?

The second one is heuristic: what would lead anyone to discover such a transform in the course of solving a problem?

(My hope is to find a unified treatment of the subject that simultaneously addresses both of these questions.)

Let me give a finite-dimensional example. Suppose we have a $2\times2$ matrix $A$ and we want to compute $A^{1000}$. Direct approach would not be very wise. However, if we first diagonalize $A$ as $PA_dP^{-1}$ (i.e. rotate the basis by $P$), the calculation becomes much easier: the answer is given by $PA_d^{1000}P^{-1}$ and computing powers of diagonal matrix is a very simple task.
A somewhat analogous infinite-dimensional example would be the solution of the heat equation $u_t=u_{xx}$ using Fourier transform $u(x,t)\rightarrow \hat{u}(\omega,t)$. The point is that in the Fourier basis the operator $\partial_{xx}$ becomes diagonal: it simply multiplies $\hat{u}(\omega,t)$ by $-\omega^2$. Therefore, in the new basis, our partial differential equation simplifies and becomes ordinary differential equation.
In general, the existence of a transform adapted to a particular problem is related to its symmetry. The new basis functions are chosen to be eigenfunctions of the symmetry generators. For instance, in the above PDE example we had translation symmetry with the generator $T=-i\partial_x$. In the same way, e.g. Mellin transform is related to scaling symmetry, etc.