Consider a function $f(x)$ and its Fourier Transform $\tilde{f}(k)$ given by $$ \tilde{f}(k) = \int_\mathbb{R}\!\!\!dx\; e^{-ikx}f(x). $$ Now, lets have the coordinate transform $\xi = \tau(x)$ and, thus, we have the Fourier Transform $\tilde{f}(\kappa)$ of $f(\xi)=f(\tau(x))$ with a new coordinate $\kappa$.
Is there a way to compute $\tilde{f}(\kappa)$ from a given $\tilde{f}(k)$ and coordinate transform $\xi = \tau(x)$?
Does the coordinate transform $k \rightarrow \kappa$ exist at all?
Thanks and regards.
There is no such coordinate transform in general: If $\tau(x) = x-x_0$ then the Fourier transform of $f\circ \tau$ is the Fourier transform of $f$ multiplied by a complex exponential.
In the case of dilation $\tau(x) = Ax$ (with invertible $A$) there is indeed a coordinate transform but this seems to be a rare exception.