Gabor frames over irregular lattices
Given a Hilbert space \(H\), a sequence \(\{x_n\}\subset H\) is a frame if for all \(y \in H\): \(A\|y\|^2 \leq \sum_{n} |\langle y, x_n \rangle|^2 \leq B\|y\|^2\). In applications, frames of a special form in \(L^2({\mathbb R})\) are important: the so-called Gabor frames or Weyl-Heisenberg frames. These are the frames generated by translations and modulations of a single function \(g \in L^2({\mathbb R})\): if we let \(T_a(g)(x) = g(x-a)\) and \(E_b(g)(x) = e^{2\pi i bx} g(x)\), a Gabor frame associated with a discrete set \(\Lambda \subset {\mathbb R}^2\) is a frame of the form \(\{E_mT_n(g): (m,n) \in \Lambda\}\). In this paper the authors study semi-irregular Gabor frames, i.e., frames of the special form \(\{E_{mb} T_{y_n}(g): m\in {\mathbb Z}, y_n \in [na, (n+1)a]\}\), for some \(a, b >0\). If \(g\) belongs to the Wiener space \(W(L^\infty, l^1)\), i.e., it satisfies the condition \(\sum_n \|T_{nc}(g) \chi_{[0,c)}\|_\infty < \infty\), for some \(c>0\), then a characterization of semi-irregular Gabor frames is possible. The main result of this paper states that the existence of constants \(a_0, b_0, A>0\) so that \(\{E_{mb} T_{y_n}(g):m,n\}\) is a frame with the lower frame bound \(A\) for all choices of \(0<b\leq b_0\) and \(y_n \in [na, (n+1)a]\) is equivalent to \(g\) being bounded below on some interval in \(\mathbb R\).
scientific article; zbMATH DE number 1896707