Characterizing the Fourier transform by its properties

TL;DR

This paper explores various unique properties that characterize the Fourier transform across different mathematical contexts, including convolution, differentiation, and shift invariance, and extends the discussion to other transforms like the Hankel transform.

Contribution

It provides new characterizations of the Fourier transform using properties like time differentiation and shift invariance, and simplifies existing proofs for convolution characterization on compact groups.

Findings

Fourier transform is characterized by its convolution property.

Differentiation and shift invariance can also characterize Fourier transforms.

Hankel transform can be characterized by a Bessel-type differential property.

Abstract

It is common knowledge that the Fourier transform enjoys the convolution property, i.e., it turns convolution in the time domain into multiplication in the frequency domain. It is probably less known that this property characterizes the Fourier transform amongst all linear and bounded operators $T:L^1 \longrightarrow C^b.$ Thus, a natural question arises: are there other features characterizing Fourier transform besides the convolution property? We answer this query in the affirmative by investigating the time differentiation property and its discrete counterpart, used to characterize discrete-time Fourier transform. Next, we move on to locally compact abelian groups, where differentiation becomes meaningless, but the Fourier transform can be characterized via time shifts. The penultimate section of the paper returns to the convolution characterization, this time in the context of compact (not necessarily abelian) groups. We demonstrate that the proof existing in the literature can be greatly simplified. Lastly, we hint at the possibility of other transforms being characterized by their properties and demonstrate that the Hankel transform may be characterized by a Bessel-type differential property.