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Colloquium Mathematicum

Uncertainty principle for Weyl transform and Fourier–Wigner transform

Amit Samanta, Santanu Sarkar

Published Jan 1, 2023• 1 citation
Publisher

Abstract

We prove an uncertainty type principle for finite linear combinations of Fourier-Wigner transforms.This is equivalent to proving a finite rank theorem for the Weyl transform.
DOI
10.4064/cm9131-8-2023

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The Fourier–Wigner transform of two functions is first defined and is given an alternative expression. The Wigner transform is then defined to be the Fourier transform of the Fourier–Wigner transform. The Moyal identity, which is an analog of the Plancherel formula, is then proved for the Wigner transform. The Weyl transform with a given symbol is then defined in terms of the Wigner transform. Another formula for the Weyl transform with a given symbol is then proved and used to establish the fact that the product of two Weyl transforms is also a Weyl transform with symbol given by a twisted convolution of the two given symbols.
Some properties of the qqq-Fourier-sine transform are studied and qqq-analogues of the Heisenberg uncertainty principle is derived for the qqq-Fourier-cosine transform studied in \cite{FB} and for the qqq-Fourier-sine transform.