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In this paper we elaborate upon the investigation initiated in [3] of typical and distinctive properties of the Fourier transform (FT), in particular the crucial role played by the Howe dual pair (O(m), sl(2)). We prove in detail a result on the unique characterization of the FT making extensive use of a representation of the Lie algebra sl(2). As an example, we consider the case m = 1. We refer to [3] for a detailed study involving the derivation of a class of operators portraying FT symmetry properties.

Unique characterization of the Fourier transform in the framework of representation theory