Abstract

In this paper are studied the transforms of operators and functions by the exponential operators , , , , , , , , , , , where without using integration. This study is facilitated by the revelation that all relations between a couple of dual operators  obeying the condition  are invariant under substituting with any another dual couple.  Compositions and decompositions of the exponential operators , making them groups are obtained. 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 integration.

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