We formulate and prove a version of Paley-Wiener theorem for the inverse Fourier transforms on noncompact Riemannian symmetric spaces and Heisenberg groups. The main ingredient in the proof is the Gutzmer’s formula.
"A Paley-Wiener theorem for the inverse Fourier transform on some homogeneous spaces." Hiroshima Math. J. 37 (2) 145 - 159, July 2007. https://doi.org/10.32917/hmj/1187916316