Shift-invariant Spaces with Countably Many Mutually Orthogonal Generators on the Heisenberg group

Let $E(\mathscr{A})$ denote the shift-invariant space associated with a countable family $\mathscr{A}$ of functions in $L^{2}(\mathbb{H}^{n})$ with mutually orthogonal generators, where $\mathbb{H}^{n}$ denotes the Heisenberg group. The characterizations for the collection $E(\mathscr{A})$ to be orthonormal, Bessel sequence, Parseval frame and so on are obtained in terms of the group Fourier transform of the Heisenberg group. These results are derived using such type of results which were proved for twisted shift-invariant spaces and characterized in terms of Weyl transform. In the last section of the paper, some results on oblique dual of the left translates of a single function $\varphi$ is discussed in the context of principal shift-invariant space $V(\varphi)$.