Bimodal Wilson systems in $L^2(\mathbb R)$

Abstract

Given a window $ϕ\in L^2(\mathbb R),$ and lattice parameters $α, β>0,$ we introduce a bimodal Wilson system $\mathcal{W}(ϕ, α, β)$ consisting of linear combinations of at most two elements from an associated Gabor $\mathcal{G}(ϕ, α, β)$. For a class of window functions $ϕ,$ we show that the Gabor system $\mathcal{G}(ϕ, α, β)$ is a tight frame of redundancy $β^{-1}$ if and only if the Wilson system $\mathcal{W}(ϕ, α, β)$ is Parseval system for $L^2(\mathbb R).$ Examples of smooth rapidly decaying generators $ϕ$ are constructed. In addition, when $3\leq β^{-1}\in \mathbb N$, we prove that it is impossible to renormalize the elements of the constructed Parseval Wilson frame so as to get a well-localized orthonormal basis for $L^2(\mathbb R)$.