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.
Best Answer
From the point of view of physics, Fourier transforms are ubiquitous because they are expansions in eigenfunctions of the derivative operator - and the derivative operator is fundamental in many aspects. Just to give two examples: The derivative operator is the generator of translations (in space or time), and to learn about the natural world, it is crucial that translations are symmetry operations - how would we learn about the natural world if we couldn't reproduce experiments at different times in different places? Secondly, field theories rely on locality, i.e., degrees of freedom only interact with their immediate neighbors - this naturally leads to dynamics described by derivatives.