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

  • Do Tan Si HoChiMinh-City Physical Association, Vietnam and ULB and UEM, Belgium.
Keywords: Fourier transform, fractional order Fourier transform, differential transform, kernels of integral transforms, transforms of geometric forms

Abstract

In this paper is proven that all relations between a couple of dual operators Screenshot_94.png i.e. operators obeying the commutation relation Screenshot_101.png are invariant under substitution of Screenshot_95.png 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. 

Published
2020-07-20
How to Cite
Si, D. T. (2020). Obtaining Differential Transforms: Applications to the Case of the Fourier Transform. New Insights into Physical Science Vol.3, 84-104. Retrieved from https://stm1.bookpi.org/index.php/nips-v3/article/view/1818
Issue
New Insights into Physical Science Vol. 3
Section
Chapters