An analogue of the Hardy theorem for the Cartan motion group (Q1283164)

Let \(G\) be the Cartan motion group, which is the semidirect product of \(K\) and \(\mathfrak{p}\), in the usual terminology of semisimple Lie groups. \textit{K. Kumahara} [Proc. Jap. Acad. 50, 433-435 (1974; Zbl 0304.43010)] defined the Fourier transform on \(G\) using the representations induced from the characters of \(\mathfrak{p}\). A form of the uncertainty principle is proved for this Fourier transform, generalizing a result of Hardy [cf. \textit{H. Dym} and \textit{H. P. McKean}, Fourier series and integrals, 155-158 (1972; Zbl 0242.42001)].