Windowed linear canonical transform: its relation to windowed Fourier transform and uncertainty principles

The windowed linear canonical transform is a natural extension of the classical windowed Fourier transform using the linear canonical transform. In the current work, we first remind the reader about the relation between the windowed linear canonical transform and windowed Fourier transform. It is shown that useful relation enables us to provide different proofs of some properties of the windowed linear canonical transform, such as the orthogonality relation, inversion theorem, and complex conjugation. Lastly, we demonstrate some new results concerning several generalizations of the uncertainty principles associated with this transformation.

Uncertainty principles associated with multi-dimensional linear canonical transform

The main objective of this paper is to establish Heisenberg’s and Beckner’s uncertainty principles associated with multi-dimensional linear canonical transform.

Uncertainty Principles for Heisenberg Motion Group

In this article, we will recall the main properties of the Fourier transform on the Heisenberg motion group G = ℍ n ⋊ K , where K = U n and ℍ n = ℂ n × ℝ denote the Heisenberg group. Then, we will present some uncertainty principles associated to this transform as Beurling, Hardy, and Gelfand-Shilov.

Uncertainty principles for the windowed offset linear canonical transform

The windowed offset linear canonical transform (WOLCT) can be identified as a generalization of the windowed linear canonical transform (WLCT). In this paper, we generalize several different uncertainty principles for the WOLCT, including Heisenberg uncertainty principle, Hardy’s uncertainty principle, Donoho–Stark’s uncertainty principle and Nazarov’s uncertainty principle. Finally, as application analogues of the Poisson summation formula and sampling formulas are given.

Localization Operators and Uncertainty Principles for the Hankel Wavelet Transform

The aim of this paper is to prove some uncertainty inequalities for the continuous Hankel wavelet transform, and study the localization operator associated to this transformation.