Characterizing the Fourier transform by its properties (Q6568717)

The fundamental aim of this work is to characterize Fourier transforms using novel approaches different from the established methodologies. This study concentrates on the characterization of Fourier transforms using a variety of properties, in addition to the convolution property that has been employed in the past. Within this framework, it is demonstrated that, under certain circumstances, the integral transforms from the Lebesgue space \(L^{1}\) into the complex-valued continuous and bounded function space \(C^{b}\) correspond to Fourier transforms. To begin, it is demonstrated that integral transforms defined from \(L^{1}\left( \mathbb{R} \right) \) into \(C^{b}\left( \mathbb{R} \right) \) are equivalent to Fourier transforms if they satisfy the Dirac delta property and the time differentiation property. Afterward, a characterization for the discrete case is defined by substituting \(\mathbb{R}\) with \(\mathbb{Z}\) and applying a suitable boundary condition according to the difference property. In the more general case of characterization for locally compact abelian groups \(G\), the time shift property and the Dirac delta property are employed to demonstrate that the integral transforms defined from the \(L^{1}\left( G\right)\) space into the \(C^{b}\left( \widehat{G}\right)\) space correspond to Fourier transforms. Then, a proof is given in the paper, where the characterization of Fourier transforms in compact groups \(G\) by the convolution is done more directly, and this proof is based on representation theory. Finally, it is demonstrated that Bessel's differential property can also be used to characterize the Hankel transform, as an additional integral operator. In light of the study's findings, the new characterizations of the Fourier transform as an integral transform and the demonstration in the final section that the Hankel transform can also be characterized by an appropriate property provide insight for future research.