Obtaining Differential Transforms: Applications to the Case of the Fourier Transform

In this paper is proven that all relations between a couple of dual operators ( A,B ) i.e. operators
obeying the commutation relation A,B I are invariant under substitution of ( A,B )with any another
dual couple.
From this property are obtained many differential operators realizing transformations in space and
phase space such as translation, dilatation, hyperbolic, … , fractional order Fourier transforms and
Fourier transform itself. Transforms of arbitrary functions and operators and geometric forms by these
differential operators are given.
The kernel of the integral transform associated with a differential transform is found. As case study the
differential Fourier transform is highlighted in order to see how it is possible to get in a concise
manner the known properties of the Fourier transform without doing integrations.