Characterizing the Fourier Transform by Phase and Amplitude Preserving Maps

It is shown that under the Fourier transform F: L 2 (R) → L 2 (R) of an even or odd function φ: R→ C, the invariance of | φ| leads to the invariance or minus invariance of arg φ, and that the inverse holds in an analogous way. Such a relation between the phase and amplitude does not necessarily hold for other integral transforms. This implies that the Fourier transform can be characterized by the phase and amplitude preserving maps. © 2017, Springer (India) Private Ltd.