A Study of Oblique Dual of a System of Left Translates on the Heisenberg Group

Abstract

Let $E(\mathscr {A})$ denote the collection of left-translates by integers of mutually orthogonal generators $\varphi _{j}$ in $L^{2}(\mathbb {H}^{n})$. Given two Bessel sequences $E(\mathscr {A})$ and $E(\mathscr {A}^{\dagger })$, we obtain a necessary and sufficient condition so that $E(\mathscr {A}^{\dagger })$ becomes an oblique dual of $E(\mathscr {A})$. Further for $\varphi \in L^{2}(\mathbb {H}^{n})$, we prove under certain additional assumptions that the canonical dual frame is the only oblique dual which has the same structure as that of $\{L_{(2k,l,m)}\varphi :(k,l,m)\in \mathbb {Z}^{2n+1}\}$.