Discrete Fourier transform in a multidimensional space

I want to measure the frequencies at which a point oscillates in a multidimensional space, let's take the example of a point on a 2d-surface. For now, I naïvely split the signal in two, along the cartesian coordinates, and compute the two FFTs separately. This approach could work but seems to be arbitrary and may loose relevant information. Is there a canonical way to perform the frequency analysis in multidimensional space ?

It is unrelated to 2D FFT. I take a vector of size 2*T and 2D FFT is concerned by plain T^2 data, like images.

Here is an example where halving the signal along the polar coordinates seems better than along the cartesian coordinates.

The data used, a point evolving in 2D,

And here is the FFTs for

1. The cartesian coordinates x and y (a bit noisy)
2. The polar coordinates theta and rho (very clear)