Bol. Soc. Mat. Mex. DOI 10.1007/s40590-017-0187-1 ORIGINAL ARTICLE Convolution with the linear canonical Hankel transformation Tanuj Kumar 1 · Akhilesh Prasad 1 Received: 29 June 2017 / Accepted: 4 October 2017 © Sociedad Matemática Mexicana 2017 Abstract In this work, we introduce translation and convolution for linear canoni- cal Hankel transformations and studied some inequalities. For the particular values of linear canonical Hankel transformation (i.e., for Hankel–Clifford transformation), we investigate linear time-invariant filters. Furthermore, some applications of linear canonical Hankel transformation to a generalized non-linear parabolic equation and a canonical convolution integral equation are given. Keywords Convolution · Linear canonical transformation · Linear time-invariant filter · Wavelets · Fredholm integral equation Mathematics Subject Classification 44A35 · 65R10 · 65R20 1 Introduction The theory of linear canonical transformation (LCT) was motivated by the work of two different projects by Collins Jr. [4] on the field of paraxial optics; on the other hand, M. Moshinsky and C. Quesne [14] in the field of nuclear physics in the early 70s. The LCT is a four parameter class of linear integral transformation for studying the behavior of many useful transformations and system responses in physics and engineering, in general. Therefore, LCT is found as a powerful mathematical tool in many fields of B Akhilesh Prasad apr_bhu@yahoo.com Tanuj Kumar tanujdimri067@gmail.com 1 Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad 826004, India