Space invariant operations

The Fourier transform of a shifted object is changed from the Fourier transform of an unshifted object only by a spatially uniform phase factor that carries the shift information. Therefore any operation in the Fourier transform domain operates in parallel on all parts of the input and is said to be space invariant.

This is a property of convolution – an act a Fourier transform performs. For more detail on shift invariance, see

http://www.ph.tn.tudelft.nl/Courses/FIP/noframes/fip-Convolut-2.html