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

Abstract

Given a window $\varphi \in L^2(\mathbb{R}),$ and lattice parameters $\alpha ,\beta >0,$ we introduce a bimodal Wilson system $\mathcal{W}(\varphi ,\alpha ,\beta )$ consisting of linear combinations of at most two elements from an associated Gabor $\mathcal{G}(\varphi ,\alpha ,\beta )$. For a class of window functions $\varphi ,$ we show that the Gabor system $\mathcal{G}(\varphi ,\alpha ,\beta )$ is a tight frame of redundancy ${\beta }^{-1}$ if and only if the Wilson system $\mathcal{W}(\varphi ,\alpha ,\beta )$ is Parseval system for $L^2(\mathbb{R}).$ Examples of smooth rapidly decaying generators $\varphi$ are constructed. In addition, when $3\le {\beta }^{-1}\in \mathbb{N}$, 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 $L^2(\mathbb{R})$.

arXiv comment25 pages