# On bicomplex Fourier--Wigner transforms

## TL;DR

This paper explores bicomplex Fourier--Wigner transforms, establishing their fundamental properties, range characterizations, and constructing an orthogonal basis using bicomplex polyanalytic functions, thus extending classical Fourier analysis into bicomplex spaces.

## Contribution

It introduces bicomplex Fourier--Wigner transforms, analyzes their properties, and constructs a bicomplex-valued orthogonal basis, advancing the understanding of bicomplex functional analysis.

## Key findings

- Established Moyal's identity for bicomplex transforms
- Characterized the range of bicomplex Fourier--Wigner transforms
- Constructed an orthogonal basis using bicomplex polyanalytic Hermite functions

## Abstract

We consider the $1$- and $2$-d bicomplex analogs of the classical Fourier--Wigner transform. Their basic properties, including Moyal's identity and characterization of their ranges giving rise to new bicomplex--polyanalytic functional spaces are discussed. Particular case of special window is also considered. An orthogonal basis for the space of bicomplex--valued square integrable functions on the bicomplex numbers is constructed by means of the polyanalytic complex Hermite functions.