A theorem of the Wiener—Tauberian type forL 1(H n)

The Heisenberg motion groupHM(n), which is a semi-direct product of the Heisenberg group Hⁿ and the unitary group U(n), acts on Hⁿ in a natural way. Here we prove a Wiener-Tauberian theorem for L¹ (Hⁿ) with this HM(n)-action on Hⁿ i.e. we give conditions on the “group theoretic” Fourier transform of a functionf in L¹ (Hⁿ) in order that the linear span of^gf : g∈HM(n) is dense in L¹(Hⁿ), where^gf(z, t) =f(g·(z, t)), forg ∈ HM(n), (z,t)∈Hⁿ.