A characterization theorem for the Fourier transform on ℝn
In this paper we develop a characterization of the Fourier transform as a continuous operator from L2(Rn) to L2(Rn) behaving like the Fourier transform under rotations, dilation, and translations. This gives a new approach to a result of A. Hertle  based on the Hecke-Bochner identities.
Bray, William O. "A characterization theorem for the Fourier transform on ℝn." Journal of mathematical analysis and applications 134, no. 1 (1988): 141-150.
Journal of Mathematical Analysis and Applications