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I'm familiar with the notion of Fourier transform in the context of $\mathbb{R}^n$ and more generally, locally compact abelian groups. However recently I came across the Fourier transform acting as follows: $M$ is a compact manifold, $x \in M$ and $T_xM,T^*_xM$ are tangent and cotangent spaces at $x$. Also $E$ is a given vector bundle over $M$. Then the Fourier transform should act $\mathcal{F}_x:\Gamma^{\infty}(T_xM,T_xM \times E_x) \to \Gamma^{\infty}(T^*_xM,T_xM \times E_x)$

Any idea how such Fourier transform should be defined?

Here $T_xM \times E_x$ should be viewed as a trivial vector bundle over $T_xM$ with fiber $E_x$. However I suspect that $\mathcal{F}_x$ should be defined somehow consistently when $x$ varies.

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I have never encountered the situation you are describing, but the following seems to make sense.

You are actually looking for a machine that eats a function $f:T_xM\to E_x$ and spits out a function $\widehat{f}:T^*_xM\to E_x.$ Imitating the classical Fourier transform, I would try $$\widehat{f}(\varphi)=\int_{T_xM}f(v)e^{-2i\pi\varphi(v)}dv.$$ Note that this is completely free of choices such as coordinates on $M$ or a trivialization of $E$.

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    $\begingroup$ ... It does need a volume form $dv$ on $T_xM$, though? $\endgroup$ Commented Jan 17, 2019 at 18:42

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