Bendlet transforms: a mathematical perspective. Issue 9 (2nd September 2022) - Shah, Firdous A.; Tantary, Azhar Y.; Lone, Waseem Z.
Abstract : Recently, Lessig et al. [Appl Comput Harmon Anal. 2019;46:384–399] introduced the notion of bendlets; a shearlet-like system that is based on anisotropic scaling, translation, shearing and bending of a compactly supported generator. The theoretical framework of the bendlet transform is yet to be explored exclusively. Taking this opportunity, our aim is to investigate the mathematical properties of the bendlet transform such as the Rayleigh theorem, inversion formula, characterization of range and the pointwise convergence of the inversion formula. In continuation, we obtain the Heisenberg-type uncertainty principle and the Pitt's inequality for the bendlet transform. Subsequently, we employ the bendlet transform for obtaining the solutions of the wave and Laplace equations. Finally, we introduce the notion of quaternionic bendlet transform and also study its fundamental properties.