In order to understand Fourier transforms in more detail, I wonder how to define the Fourier transform on affine spaces. In other words: How to define the FT while maintaining a strict separation between points and vectors.
Given an affine space $(A,V,-)$ and a function $f: A \to \mathbb{R}$, what is the Fourier transform of $f$?
$\hat{f}(\mathbf{k}) = \int_A{\exp(-2\pi \mathbb{i} \mathbf{k} \cdot \mathbf{x} ) f(\mathbf{x}) }$ is malformed because there is no such thing as a scalar product of points.
It is easy and non-insightful to arbitrarily choose an origin $\mathbf{0} \in A$ and simply define the Fourier transformation on $V$. One can then show that the Fourier transformation is independent of the choice of $\mathbf{0}$, up to a global phase:
$\hat{f}(\vec{k}) = \int_V{\exp(-2\pi \mathbb{i} \vec{k} \cdot \vec{v} ) f(\mathbf{0} + \vec{v}) }$.
Is there a more insightful, abstract definition of the Fourier transform for affine spaces? If not, is there an intrinsic reason, why the FT requires the choice of an origin?
PS: I tried to find the answer for general smooth manifolds, but as it seems in the case of curved manifolds, there are more urgent problems than the difference between points and vectors: Does a Fourier transformation on a (pseudo-)Riemannian manifold make sense?