Coordinate conversion

[erdemath]

A 17th century french author named Nicolas Boileau said

Ce qui se conçoit bien s'énonce clairement, et les mots pour le dire arrivent aisément.

Unfortunately people don't speak such a great french today, so that automatic translators fail to render this in english, but it means approximately

Whatever is well conceived is clearly said, And the words to say it flow with ease.

May I suggest that the underlying problem is not well enough conceived? I hope you'll find a better helper than me...

I wish we had some auto-compiler here, like Latex. Let us consider the usual Euclidean domains (i.e., finite dimensional Hilbert spaces) \Omega_1 and
\Omega_2. Now, we define the following mapping

\Omega_1 --> \Omega_2
x --> f(x):=y

Which means, for any x \in \Omega_1, y := f(x) \in \Omega_2 . Here, the domains \Omega_1 and \Omega_2 are in the Euclidean space.
What I am questioning, what is the Fourier domain of \Omega_1 ? Is it simply the Fourier projection of the entire domain \Omega_1? Or, would it
still be fine if we consider polar coordinates as a conversion of \Omega_1?