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

Question

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.)

Answer 1

The essential idea of many transforms is to change the basis in the space of functions with the hope that in the new basis the problem will simplify.

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.

Answer 2

The closest thing to a general theory that leads to MANY of the above (though not all) is Sturm–Liouville theory. Basically, many of these transforms have come about from the study of physical phenomena via linear differential equations, where, as previous answers have noted, specific transforms diagonalize the differential operator. It turns out that MANY physical phenomena of interest obey second order differential equations of the Sturm-Liouville type. The same logic really applies for other differential equations (or difference equations in the case of the z-transform). Once you know what functions fundamentally solve a linear differential equation, you want to make up more functions that solve the problem by an integral or sum over these fundamental solutions; this idea leads to many of the transform above. Spectral theory of operators and ideas from Hilbert spaces generalize this for higher order operators. Each one of these equation types naturally appear in physical models of the world. I'll outline some of the differential equations I mean, the associated transform, and the physical applications in which they came about.

1. Linear constant coefficient ODEs with zero boundary conditions before t=0. The function $e^{st}$ solves these for some values of $s$. Superposing these leads to the Laplace transform. Mellin is closely related. Equations model kinematics, circuits.

2. Linear constant coefficient ODEs or PDEs in unbounded domains. Plane waves $e^{jkr}$ in multiple dimensions solve these for a continuum of $k$ values. Superposing these leads to the (multidimensional) Fourier transform. In bounded domains some of these dimensions reduce to summations instead of integrals. In certain cylindrical symmetry the solutions are Bessel and Hankel functions, reducing to the Hankel transform. Equations model wave mechanics, heat conduction, potential theory, etc.

3. Linear constant coefficient difference equations in the variable $n$. The function $z^n$ will solve these equations for some particular values of $z$. Superposing these leads to the z transform. Linear recurrences appear in the math of sequences and series, digital filters, generating functions in probability.

Some of the methods you mention are not from this family of naturally arising from differential equations, namely the Legendre and Hilbert transforms. The Hilbert has a similar form of a linear integral transform, and could be considered unified with the rest. The Legendre transform is something else entirely however.