Suppose we have some Schwartz function $f$. Denote its Fourier transform $\widehat{f}$. Now, I am trying to find a way to write the ratio
$$\frac{\widehat{f}(\eta-\xi_0)}{\widehat{f}(\eta-\xi_0-\Delta)}=\frac{\int_{-\infty}^\infty e^{2\pi i\xi_0 x} f(x) e^{-2\pi i \eta x} dx}{\int_{-\infty}^\infty e^{2\pi i(\xi_0+\Delta) x} f(x) e^{-2\pi i \eta x} dx}$$
as (ideally) a single invertible function of $\eta$. However, this likely is not possible and I am looking for any ideas/insights about the ratio of two frequency shifted Fourier transforms such as this.