variables of Fourier analysis - how to prove their relations

Question

Fourier transform: $\hat{f}(\xi) = \int_{-\infty}^{\infty} f(t)\ e^{- 2\pi i t \xi}\,dx$ where $t$ can be time and $\xi$ can be frequency. So, the question is how do we prove that $t$ and $\xi$ can in fact be time-frequency combination?

Answer 1

Multiply them and check that $t\xi\in \Bbb R$.