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Is there a category theoretic definition for the Fourier transform using only its universality properties? I am not looking for the most general definition -- one that works only in some special settings will do. I am looking for a simple definition that will make precise my (possibly incorrect) intuition that Fourier transforms are in some sense extremal among unitary transforms.

Here is another non category-theoretic way to ask this question, which may or may not be equivalent: Give a "natural" optimization problem on the space of unitary transforms whose solution turns out to be the Fourier transform.

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    $\begingroup$ I assume you're not interested in something like Pontryagin duality. If that's the case, you might want to edit your question/title accordingly. $\endgroup$
    – arsmath
    Oct 15, 2010 at 21:21
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    $\begingroup$ What universality properties? What categories do you have in mind? $\endgroup$
    – Yemon Choi
    Oct 16, 2010 at 4:31
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    $\begingroup$ -1: This question needs to be made more specific. Are you looking for a rule that assigns to each locally compact group $A$ a special function on $Isom(L^2(A), L^2(A^\vee))$ that is minimized by the Fourier transform, in a way that is compatible with homomorphisms? $\endgroup$
    – S. Carnahan ♦
    Oct 16, 2010 at 6:04
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    $\begingroup$ +1 Often the point of asking a question is not so much to get an answer, but rather to be shown how to improve the question. Fortunately, the above comment does suggests such an improvement, among other things. $\endgroup$
    – Chris Brav
    Oct 16, 2010 at 8:52
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    $\begingroup$ @Chris: I don't agree: at least, I do not think that it's a good kind of question to ask on MO. It's what research supervisors (and teachers) should be there for. Asking vague questions in the hope that someone will suggest the correct version, seems too much like asking other people to do the work, at least on MO. $\endgroup$
    – Yemon Choi
    Oct 17, 2010 at 0:24

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For me, the "traditional Fourier Transform" is a change of basis of the algebra of functions from a group to some chosen field: from the canonical basis to something sometimes called the Fourier Basis. Because the Transform is constructed using the representation theory of the group, it has "natural" generalisations to objects with "similar" representation theory, e.g. it is defined for Hopf algebras.

I always think of the FT as this kind of duality. The Fourier Basis have lot's of interesting properties, but I have not seen a definition of it using extremals. It would be very interesting to see that.

A good start would be if someone gives an answer for finite abelian groups and finite non-abelian groups. Though "non-abelian" groups have a notion of FT, it is not uniquely defined and it is hard to work with it. An "extremal" condition would be enlightening.

Update 14th Dec 2011

Sorry that this comes several months later, but I found that there is "a way" to define the quantum Fourier transform for abelian finite groups using an extremal argument. This argument comes from reference [1] where the Fourier transform is studied as a tool to design measurements in Quantum Computation and proven to be optimal to solve the abelian hidden subgroup problem. Unfortunately, this property does not hold for non-abelian quantum Fourier transforms.

More concretely, what it is proven in [1] is the following (all definitions I use are defined in this paper):

Consider the hidden subgroup problem defined for an abelian group $G$, where the hidden subgroup $H$ is chosen uniformly at random from all subgroups of $G$. Given $n$ tensored random coset states (cosets of $H$), then the measurement that maximises the probability of correctly identifying the subgroup $H$ is the following:

  1. Start on a random coset-state $|x+H\rangle$ for unknown $H$ which is just a uniform quantum superposition over the elements of the coset $x+H$. Cf. [1] for details on how to create these quantum states.
  2. Apply the abelian quantum Fourier transform of $G$ on this state.
  3. Perform a projective measurement.

Taking several outcomes of the above procedure one obtains a generating set of the orthogonal group $H^\perp$ from which the original subgroup $H$ can be recovered solving a system of linear modular equations [2]. As far as I know these "orthogonal subgroups" are sometimes called orthogonal complements in Mathematics.

To sum up, they key ingredient of the above quantum algorithm is the abelian Fourier transform which is used to implement a quantum measurement to solve the hidden subgroup problem since it maximises the probability of distinguishing the hidden subgroup. In [1] it is shown that the abelian quantum Fourier transform arises as an optimal POVM which is the solution of a semidefinite program. I guess that maybe you could adopt this kind of extremal property as a definition of the Fourier transform for finite abelian groups. Note: it is not clear to me that the optimal POVM found in [1] is unique (up to permutations).

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  • $\begingroup$ I know I did not answer the question, but I think that even for simpler mathematical objects that the ones you ask it is not obvious that FT can be defined with an optimization problem. $\endgroup$
    – Juan Bermejo Vega
    Jun 13, 2011 at 11:22

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