Finding duality for Riesz bases of exponentials on multi-tiles

It is known that if $Ω\subset \mathbb{R}^{d}$ belongs to a class of multi-tiling domains when translated by a lattice $Λ$, there exists a Riesz basis of exponentials for $L^{2}(Ω)$ constructed using $k$ translates of the dual lattice $Λ^*$. In this paper, we give an explicit construction of the corresponding biorthogonal dual Riesz basis. We also extend the iterative reconstruction algorithm introduced in prior work to this setting.