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 multiplicity of such transforms. 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.)

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A lot of these transforms are just linear transformations on vector spaces of functions. The Fourier transform, for example, is just an orthogonal transformation on $L^2(\mathbb R)$, which I think of as changing the basis from a bunch of delta functions to a bunch of sinusoids. The Legendre transform is kind of interesting because it turns out to be the exact analogue of the Fourier transform if you replace the ring $(\mathbb R,+,\times)$ with the "tropical semiring" $(\mathbb R\cup\{-\infty\},\max,+)$. –  Rahul Jun 8 '13 at 14:11
en.wikipedia.org/wiki/… with various twistings –  yoyo Jun 8 '13 at 14:16
+1 This is a beautiful question. –  Ben Blum-Smith Jun 8 '13 at 15:07
You might be interested in things like the FHA cycle... –  Ｊ. Ｍ. Jun 8 '13 at 15:51
I find this a little unsatisfying - I'd be interested in a more algebraic (as opposed to linear algebraic) unifying picture building off of the fact that the Mellin transform is the multiplicative analogue of the Laplace/Fourier transform, and the Legendre transform is the tropical analogue of the Fourier transform. I don't know if such a picture exists, but if it does, I would be very interested. (Or would everything in my picture just being a special case of a representation-theoretic picture?) –  Davidac897 Jun 27 '13 at 15:52
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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.

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How exactly does the Fourier transform change the basis on $L^2(\mathbb R)$? What are the 'before' and 'after' bases? –  Potato Sep 1 '13 at 7:01
@Potato The "before" basis is composed of "functions" $\delta(x-y)$, where $x$ is the argument and $y$ is a parameter. I.e. we represent $f(x)$ as $\int_{\mathbb{R}}f(y)\delta(x-y)dy$ and look at $f(y)$ as being coordinates of $f(x)$. The "after" basis is formed by the functions $e^{i\omega x}$, where $x$ is again the argument and $\omega$ is a parameter. –  O.L. Sep 1 '13 at 9:17
I'm confused because neither of those functions are in $L^2$. Do you know of a good reference where I can learn more? (I'm already familiar with basic Fourier theory, just not this perspective.) –  Potato Sep 1 '13 at 16:45
@Potato I just attempted to give an informal explanation and I am not sure whether it can be made mathematically rigorous. However, this is how Fourier transformation is usually understood by physicists. If $f(x)$ is interpreted as a wave function, the old and new basis correspond to its coordinate and momentum representation. (The basis of plane waves $e^{ipx}$ diagonalizes the momentum operator $\hat{p}=-i\partial_x$.) I will think about the reference, but nothing comes to mind immediately. –  O.L. Sep 1 '13 at 17:08
After thinking about it some more, it makes a more sense. Thank you for the explanation. –  Potato Sep 1 '13 at 17:14
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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.

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