Multidimensional Quaternionic Gabor Transforms

Abstract

In this paper, we extend the Gabor transform to the quaternion valued functions on \({\mathbb{R}^{d}}\) in two different ways, where \({d\in \mathbb{N}}\) is arbitrary. We prove that the quaternionic Gabor transforms satisfy the properties including Parseval relation, inversion formula, linearity and uncertainity principle. We also present an extension of a quaternionic Gabor transform to Boehmians.