Abstract
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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
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Thus, a natural question arises: are there other features characterizing the Fourier
transform besides convolution property? We provide an affirmative answer 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 with the aid of
representation theory techniques. 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.
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Keywords
Fourier transform, convolution property, representation
theory on compact groups, Hankel transform
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Mathematical Subject Classification
Primary: 42A38, 43A25, 43A30
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Milestones
Received: 20 February 2024
Revised: 22 June 2024
Accepted: 23 June 2024
Published: 6 July 2024
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| © 2024 MSP (Mathematical Sciences
Publishers). Distributed under the Creative Commons
Attribution License 4.0 (CC BY). |
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