# Explaining Mukai-Fourier transforms physically

A core concept in mathematics, engineering, and physics is the Fourier Transform (FT) and its many variants (Generalized Fourier Series, Green's Function, Pontryagin duality).

The basic algorithm is to find dual sets of eigenvectors/eigenfunctions parametrized by a continuous (e.g., $\omega$ below) or discrete index (e.g., $n$ below), that satisfy completeness and orthogonality relations encapsulated in Dirac delta function resolutions such as that for the FT

$$\delta(x-y)= \int_{-\infty}^{\infty}\exp(i2\pi \omega x)\exp(-i2\pi \omega y)d\omega$$

giving

$$\int_{-\infty}^{\infty}f(y)\delta(x-y)dy=f(x)=\int_{-\infty}^{\infty}\exp(i2\pi \omega x)\int_{-\infty}^{\infty}f(y)\exp(-i2\pi \omega y) dy d\omega$$

$$=\int_{-\infty}^{\infty}\exp(i2\pi \omega x)\hat{f}(\omega) d\omega,$$

or that for the eigenvectors of Sturm-Liouville differential operators over finite domains

$$\delta(x-y)=\sum_{n=0}^{\infty }\Psi_n(x)\Psi_n^*(y)$$

giving

$$f(x)=\sum_{n=0}^{\infty }\Psi_n(x)\int_{a}^{b}f(y)\Psi_n^*(y) dy,$$

or Kronecker delta resolutions such as that for the associated Laguerre functions

$$\frac{(n+\alpha)!}{n!}\delta_{mn}=\int_{0}^{\infty}x^{\alpha}e^{-x}L_{n}^{\alpha}(x)L_{m}^{\alpha}(x)dx$$

giving

$$f(x)=\sum_{n=0}^{\infty }\frac{n!L_{n}^{\alpha}(x)}{(n+\alpha)!}\hat{f}_n$$

with

$$\hat{f}_n=\int_{0}^{\infty}x^{\alpha}e^{-x}L_{n}^{\alpha}(x)f(x)\,dx.$$

The basic "physical" operation (BPO) at work here can be regarded as destructive/constructive interference; the product at a point of the value of the function (to be resolved) with the corresponding value of an eigenfunction has a negative or positive value (or phase factor) that may sum constructively or destructively with products at other points (seen as a matched filtering or correlation by replacing $y$ with $x-z$ above). Alternatively, the BPO may be viewed as projection of vectors onto a set of orthonormal axes. In addition, if the function and operations are discretized and/or the domains restricted (in one space or its dual or both, as for the DFT) aliasing (which seems analogous to the introduction of equivalence classes) is introduced and periodicity imposed.

Can you explain the machinery behind the Mukai-Fourier transform in terms of these BPOs or close analogies?

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(Generalizing) There are two interesting narratives on the history of harmonic analysis that interweave developments in mathematical physics and mathematical analysis. For the short version, see Norbert Wiener's "The historical background of harmonic analysis" (ams.org/samplings/math-history/procsemi-wiener.pdf), and for the long version, G. Mackey's "Harmonic analysis as the exploitation of symmetry--a historical survey" (ams.org/journals/bull/1980-03-01/S0273-0979-1980-14783-7/…). Can someone extend the narrative to the Mukai-Fourier transform? – Tom Copeland Sep 28 '12 at 10:06
Apropos Scott Carnahan's answer, see pg. 23 of E. Frenkel's "Langlands Program, Trace Formulas, and their Geometrization" (arxiv.org/abs/1202.2110). – Tom Copeland Sep 30 '12 at 19:18

You can think of line bundles and skyscraper sheaves as sheaf-theoretic analogues to exponentials and delta functions, respectively. The Fourier-Mukai transform on an elliptic curve takes one type of sheaf to the other (with a homological shift that I will ignore). In higher dimension, you get some mixtures of these types, from complexes of sheaves with cohomology supported on subvarieties of positive dimension and positive codimension - you can think of these as an analogue of more general distributions. Vector bundles on an abelian variety get "orthogonally decomposed" into skyscrapers on the "frequency space", i.e., the dual abelian variety.

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Nice analogy, thanks. – Tom Copeland Sep 27 '12 at 10:24
I like it too! is great! – Dox Sep 27 '12 at 12:23
Can you deepen the analogy to include properties of the FT such as Parseval's relation or conversion of differentiation to multiplication? – Tom Copeland Sep 27 '12 at 23:24
I do not know an analogue of unitarity. The relationship between linear polynomials and directional derivatives in the usual Fourier theory can be rephrased in terms of an endomorphism on the ring of differential operators on affine space. You may be able to do something similar with the derived category of coherent sheaves on the product of the abelian variety and its dual, but I don't know a concrete answer. – S. Carnahan Sep 28 '12 at 3:59
You can't relate a "translation" (if that can be made sensible) of a skyscraper sheaf to "multiplication" by a "function" in the "dual space"? – Tom Copeland Sep 28 '12 at 9:29

I'd suggest you to check the post HERE.

As you will see, the analogy enters by thinking the pullback of the sheaf $\mathcal{F}$ on the $X$ variety, i.e. $p_1^*\mathcal{F}$, as Fourier coefficients, while the sheaf $\mathcal{P}$ on the product variety $X\times Y$ plays the role of integral kernel.

Cheers.

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That post actually inspired my post to seek clarification and to probe how deeply the analogy goes. – Tom Copeland Sep 27 '12 at 6:14