Characterization of the group Fourier transform of Hilbert space Valued functions

A remarkable theorem of S. Alesker, S. Artstein-Avidan, and V. Milman characterizes the Fourier transform on Rⁿ as, in essence, the unique transformation on the space of tempered distributions that exchanges convolution with pointwise multiplication. In this paper, we extend this perspective to the setting of the Heisenberg group , focusing on functions taking values in a separable Hilbert space . We establish a characterization of the group Fourier transform on in terms of convolution. More precisely, we characterize the group Fourier transform on L¹(,) and L²(,). In addition, we obtain a similar characterization for the group Fourier transform of -valued Schwartz class functions on the Heisenberg group.