Necessary part of Wexler–Raz-type relations for Gabor frames on L2(0,1)

Establish that for the Gabor system Gy(g, Zβ) on L2(0,1), where g ∈ S0(R) is a window and Zβ = iβZ is the regular lattice in the flat cylinder [0,1) × R, the Gabor frame property implies the existence of a dual window y ∈ S0(R) satisfying the periodized Wexler–Raz-type biorthogonality relations B^{-1}⟨y, M_n T_k g⟩_{L^2(R)} = δ_{k,0} for all integers k and n, as formulated in equation (4.18).

Background

In the classical setting L2(R), the Wexler–Raz biorthogonality relations provide necessary and sufficient conditions for Gabor frames indexed by lattices. In the present work, the authors develop an analogue on the flat cylinder by periodizing the STFT, deriving Janssen’s representation and a sufficient Wexler–Raz-type criterion for frames on L2(0,1).

However, only the sufficiency is established: the authors could not prove the necessity that a Gabor frame on L2(0,1) must admit a dual window y ∈ S0(R) obeying the periodized biorthogonality relations (4.18). Resolving this would complete the analogue of Wexler–Raz for the quasi-periodic L2(0,1) setting and clarify the structural theory of such frames.