In this paper is proven that all relations between a couple of dual operators ) B,A( i.e. operators obeying the commutation relation I B,A are invariant under substitution of ) B,A( 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.
Do Tan Si
HoChiMinh-City Physical Association, 40 Dong Khoi, Q1, TP.HCM, Vietnam and Université libre de Bruxelles and UEM, Belgium.
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