Bimodal Wilson systems in L-2(R)

Abstract

Given a window φ ∈L2(R), and lattice parameters α, β>0, we introduce a bimodal Wilson system W(φ, α, β) consisting of linear combinations of at most two elements from an associated Gabor G(φ, α, β). Fo r a class of window functions φ, we show that the Gabor system G(φ, α, β)is a tight frame of redundancy β−1if and only if the Wilson system W(φ, α, β)is Parseval system for L2(R). Examples of smooth rapidly decaying generators φare constructed. In addition, when 3 ≤β−1∈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 L2(R).