Dual pairs of Gabor frames for trigonometric generators without the partition of unity property (Q2892237)

The aim of this paper is to provide explicit constructions of dual pairs of frames having Gabor structure. To be more precise, let us denote by \(T_a\) and \(E_b\) the translation and modulation operators on \(L^2({\mathbb R})\) defined by \((T_af)(x) = f(x-a)\) and \((E_bf)(x) = e^{2\pi i b x}f(x).\) Let \(g\in L^2({\mathbb R})\) and \(a, b > 0\) be given. The system \(\{E_{mb}T_{na}g\}_{m,n\in {\mathbb Z}}\) is called a Gabor frame if there exist constants \(A,B > 0\) such that NEWLINE\[NEWLINE A||f||^2 \leq \sum_{m,n\in {\mathbb Z}}|\left<f, E_{mb}T_{na}g\right>|^2 \leq B||f||^2\;\;\forall f\in L^2({\mathbb R}).NEWLINE\]NEWLINE A typical result of the article is as follows. Let \(\eta \in {\mathbb N}, a = 1, b\in ]0, \frac{1}{5}]\) and consider the function \(g(x) = \sin^\eta(\frac{1}{3}\pi x)\chi_{[0,3]}(x).\) Then \(\{E_{mb}T_{na}g\}_{m,n\in {\mathbb Z}}\) is a Gabor frame admitting a dual Gabor frame \(\{E_{mb}T_{na}h\}_{m,n\in {\mathbb Z}}\) for some \(h\) of the form \(h(x) = (\frac{4}{3})^\eta b (Cg(x) + Dg(x+1) + Eg(x+2))\) if and only if \(\eta < 6\).NEWLINENEWLINE This is part of Theorem 2.2, which also contains a discussion of the concrete expression for \(h\) in the case \(\eta < 6.\) Frames generated by \(g(x) = \sin(\frac{\eta}{3}\pi x)\chi_{[0,3]}(x)\) are also considered.