A theorem of the Wiener-Tauberian type for \(L^ 1(H^ n)\)
Let \(H^n= C^n \times R\) denote the \(n\)-dimensional Heisenberg group and \(HM(n) =H^n \times U(n)\) the Heisenberg motion group, where \(U(n)\) is the compact group of \(n\times n\) unitary matrices with entries in \(C\). The group \(HM(n)\) acts on \(H^n\) in the following way: \((\sigma z,t) (w,s)= (\sigma w+z\), \(s+ t+1/2I_m \text{ w.z})\). The author proves a Wiener-Tauberian theorem for \(L^1(H^n)\) with this \(HM(n)\) action on \(H^n\), i.e., he provides conditions on the ``group theoretic'' Fourier transform of a function \(f\in L^1(H^n)\) in order that the linear span of \(\{g_f: g\in HM(n)\}\) is dense in \(L^1(H^n)\), where \(g_f(z,t) =f(g. (z,t))\) for \(g\in HM(n)\), \((z,t) \in H^n\).
scientific article; zbMATH DE number 1043987