# Bicomplex analogs of Segal-Bargmann and fractional Fourier transforms

## TL;DR

This paper develops bicomplex analogs of the Segal-Bargmann and fractional Fourier transforms, introducing explicit integral operators and exploring their properties within the bicomplex function space framework.

## Contribution

It introduces and analyzes bicomplex versions of the Segal-Bargmann transform and fractional Fourier transform, providing explicit formulas and their properties.

## Key findings

- Explicit integral operator connecting complex and bicomplex Bargmann spaces.
- Introduction of a bicomplex Segal-Bargmann transform and its inverse.
- Definition and exploration of bicomplex fractional Fourier transforms.

## Abstract

We consider and discuss some basic properties of the bicomplex analog of the classical Bargmann space. The explicit expression of the integral operator connecting the complex and bicomplex Bargmann spaces is also given. The corresponding bicomplex Segal--Bargmann transform is introduced and studied as well. Its explicit expression as well as the one of its inverse are then used to introduce a class of two--parameter bicomplex Fourier transforms (bicomplex fractional Fourier transform). This approach is convenient in exploring some useful properties of this bicomplex fractional Fourier transform.