TITLE: Why is the Fourier transform so ubiquitous?
QUESTION [63 upvotes]: Many operations and equivalences in mathematics arise as some sort of Fourier transform. By Fourier transform I mean the following:
Let $X$ and $Y$ be two objects of some category with products, and consider the correspondence $X \leftarrow X \times Y \to Y$. If we have some object (think sheaf, function, space etc.) $\mathcal{P}$ over $X \times Y$ and another, say $\mathcal{F}$ over $X$, assuming the existence of suitable pushpull and tensoring operations, we may obtain another object over $Y$ by pulling $\mathcal{F}$ back to the product, tensoring with $\mathcal{P}$, then pushing forward to $Y$. 
The standard example is the Fourier transform of functions on some locally compact abelian group $G$ (e.g. $\mathbb{R}$). In this case, $Y$ is the Pontryagin dual of $G$, $\mathcal{P}$ is the exponential function on the product, and pushing and pulling are given by integration and precomposition, respectively. 
We also have the Fourier-Mukai functors for coherent sheaves in algebraic geometry which provide the equivalence of coherent sheaves on dual abelian varieties. In fact, almost all interesting functors between coherent sheaves on nice enough varieties are examples of Fourier-Mukai transforms. A variation of this example also provides the geometric Langlands correspondence
$$D(Bun_T(C)) \simeq QCoh(LocSys_1(C))$$
for a torus $T$ and a curve $C$. In fact, the geometric Langlands correspondence for general reductive groups seems to also arise from such a transformation. 
By the $SYZ$ conjecture, two mirror Calabi-Yau manifolds $X$ and $Y$ are dual Lagrangian torus fibrations. As such, the conjectured equivalence
$$D(Coh(X)) \simeq Fuk(Y)$$
is morally obtained by applying a Fourier-Mukai transform that turns coherent sheaves on $X$ into Lagrangians in $Y$. 
To make things more mysterious, a lot of these examples are a result of the existence of a perfect pairing. For example, the Poincaré line bundle that provides the equivalence for coherent sheaves on dual abelian varieties $A$ and $A^*$ arises from the perfect pairing
$$A \times A^* \to B\mathbb{G}_m.$$
Similarly, the geometric Langlands correspondence for tori, as well as the GLC for the Hitchin system, arise in some sense from the self-duality of the Picard stack of the underlying curve. These examples seem to show that non degenerate quadratic forms seem to be fundamental in some very deep sense (e.g. maybe even Poincaré duality could be considered a Fourier transform).
I don’t have a precise question, but I’d like to know why we should expect Fourier transforms to be so fundamental. These transforms are also found in physics as well as many other “real-world” situations I’m even less qualified to talk about than my examples above. Nevertheless I have the sense that something deep is going on here and I’d like some explanation, even if philosophical, as to why this pattern seems to appear everywhere.

REPLY [7 votes]: A signature property of the fourier transform is that it converts convolution into multiplication. (This is crucial for real world applications such as image processing especially as in the discrete case the fourier transform can be computed very quickly using the FFT).
Another signficant property of the fourier transform is that it measures how randomly a finite set is distributed via the size of its fourier coefficients.
This is applied in Roth's proof of Szemeredi's as one can show that if all the fourier coefficients of a set are small then the set has approximately the right number of arithmetic progressions of length three. 
Similarly, counting solutions to algebraic equations over finite fields uses the fact that the values of polynomials including simple powers are randomly distributed and this can be measured using the fourier transform and exponential sum estimates.