TITLE:
From Translation to Linear and Linear Canonical Transformations
AUTHORS:
Tan Si Do
KEYWORDS:
Dual Operators, Fundamental Law of Operator Calculus, Newtonian Binomial and Translation, Linear and Linear Canonical Transforms, From Fourier to Gauss and LCTs’ Transforms
JOURNAL NAME:
Applied Mathematics,
Vol.13 No.6,
June
21,
2022
ABSTRACT: In order to obtain with simplicity the known and new properties of linear canonical transformations (LCTs), we show that any relation between a couple of operators (A,B) having commutator identical to unity, called dual couple in this work, is valuable for any other dual couple, so that from the known translation operator exp(a∂x) one may obtain the explicit form and properties of a category of linear and linear canonical transformations in 2N-phase spaces. Moreover, other forms of LCTs are also obtained in this work as so as the transforms by them of functions by integrations as so as by derivations. In this way, different kinds of LCTs such as Fast Fourier, Fourier, Laplace, Xin Ma and Rhodes, Baker-Campbell-Haussdorf, Bargman transforms are found again.