Clifford-Valued Shearlet Transforms on Cl(P,Q)-Algebras

The shearlet transform is a promising and powerful time-frequency tool for analyzing nonstationary signals. In this article, we introduce a novel integral transform coined as the Clifford-valued shearlet transform on Cl(p,q) algebras which is designed to represent Clifford-valued signals at different scales, locations, and orientations. We investigated the fundamental properties of the Clifford-valued shearlet transform including Parseval's formula, isometry, inversion formula, and characterization of range using the machinery of Clifford Fourier transforms. Moreover, we derived the pointwise convergence and homogeneous approximation properties for the proposed transform. We culminated our investigation by deriving several uncertainty principles such as the Heisenberg-Pauli-Weyl uncertainty inequality, Pitt's inequality, and logarithmic and local-type uncertainty inequalities for the Clifford-valued shearlet transform.

Automated summary

The article introduces the Clifford-valued shearlet transform for representing Clifford-valued signals and investigates its fundamental properties and uncertainty principles.