Bimodal Wilson systems in L2( 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). For a class of window functions φ, we show that the Gabor system G(φ, α, β) is a tight frame of redundancy β-1 if and only if the Wilson system W(φ, α, β) is Parseval system for L2( R). Examples of smooth rapidly decaying generators φ are 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).
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