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.

最近，Lessig 等人。[应用计算机哈蒙肛门。2019;46:384–399] 引入了弯管的概念；一种基于紧凑支撑发电机的各向异性缩放、平移、剪切和弯曲的类剪切系统。弯折变换的理论框架还有待专门探索。借此机会，我们的目标是研究弯曲变换的数学性质，例如瑞利定理、反演公式、范围的表征和反演公式的逐点收敛。接着，我们得到了海森堡型不确定性原理和 Bendlet 变换的 Pitt 不等式。随后，我们使用弯曲变换来获得波和拉普拉斯方程的解。最后，