### Transforms

#### Posted by David Corfield

I have a project afoot to gain some historical understanding on the rise of appreciation for mathematical duality. Something I need to know more about is the process whereby the duality involved in Fourier analysis came to be seen as arising through a pairing of a space with its dual, and so allowing comparison to other such dualities.

This has got me thinking once again about transforms, something we’ve discussed many times at the Café before. In as general a setting as possible, we could take some pairing

$A \times B \to C,$

and then if we have a map $g$ from $C$ to a rig $D$, we can transform certain functions from $A$ to $D$ to ones from $B$ to $D$. We would do this by forming the $D$-sum over $a$ of $D$-products of $g((a, b))$ and $f(a)$.

Usually we pick $C$ to act as a dualizing object, and then $B$ to be the dual of $A$ with respect to $C$. So in the case of the Fourier transform for locally compact abelian groups, we choose $C$ to be the circle group, and $B$ as the group of characters of $A$.

In this case, we take $D$ to be the complex numbers, and we map the circle group to the unit complex numbers. Then the transform of $f$, a complex function on $A$, is $\hat f(\chi) = \int_a \chi (a) \cdot f(a) d \mu$, integrating with respect to the Haar measure on $A$.

We might then realise that we could have chosen a larger $C$, and take it as all of $\mathbb{C}$. The maps $A$ to $\mathbb{C}$ were called ‘generalized characters’ by George Mackey in his 1948 paper, The Laplace Transform for Locally Compact Abelian Groups. So now the dual of $A$ is the product of the ordinary characters of $A$ (the complex part) and the real linear functionals (once logarithms are taken). We find that in the case of the integers, the dual is now not just the unit circle as in the case of the Fourier dual, but all of the non-zero complex numbers.

Now in this case we have $C$ and $D$ identical, and can form the Laplace transform for any locally compact abelian group. Perhaps more usually the Laplace transform is taken to be the real version of the complex Fourier transform, whereas Mackey is treating them in combination.

This reminded me of something we discussed years ago, looking at the Legendre transform as a deformation to a different rig $D$ of the Laplace transform. In the guise of the Legendre-Fenchel transformation, we have $A$ a real vector space $X$, $B$ its dual, with obvious pairing to $\mathbb{R}$. We take the rig $D$ to be $\mathbb{R}_{max}$, the reals extended by $\{ - \infty\}$, with ‘multiplication’ as $+$, ‘addition’ as $max$. This is usually extended further to takes maps into $\mathbb{R}_{max} \union \{+ \infty\}$. We can see that such a rig is at play because instead of the integral we now have $sup$, and we don’t multiply $f(a)$ with the evaluation $(a, b)$, but add (or rather subtract, there being sign conventions)

$f^{\ast} (x^{\ast}) = sup \{ \langle x^{\ast}, x \rangle - f(x)| x \in X \}.$

This has always struck me as a bit odd that we’d use something valued in $\mathbb{R}$ in the very different setting of the rig $\mathbb{R}_{max}$. It works, I guess, because we can map the reals under addition to the multiplicative structure of $\mathbb{R}_{max}$. But as all this transform business seems to be a souped-up form of matrices acting on vectors, I wonder whether there isn’t a way of taking $A$ and its dual as semimodules, that is, modules for semirings or rigs. Couldn’t the pairing of the spaces take values in $\mathbb{R}_{max}$? I need to look closely at it, but something like this may be being described in Duality and Separation Theorems in Idempotent Semimodules, see example 7.

We see transforms everywhere, such as the Radon transform for the incidence pairing for homogeneous spaces $G/H \times G/K \to \{0, 1\}$, transforming complex maps on $G/H$ to ones on $G/K$, allowing us, say, to reconstruct the value of a function at a point from the integrals of the function on lines passing through the point (see here). Here again our $C$ differs from our $D$. I’d love to understand the rationale behind such choices.

What makes for a transform that you can write whole books about? The membership pairing for a set $A$ and its powerset $P(A)$, leads to the transform of a function $f$ from $A$ to the rig of truth values (i.e. a subset $B$) being the subset of $P(A)$ formed of subsets of A which intersect with B. Not enough there to fill a tome it appears.

Up a categorical level, we have composition of profunctors $F\colon A ⇸ B$ with $G\colon 1 ⇸ A$ or $H\colon B ⇸ 1$. Do any of these get recognised as worthy transforms? What if a category like $Hilb$ were the target rather than $Set$?

## Re: Transforms

Oooh. I was about to post something about this stuff. Watch this space.