Representations of Gabor frame operators (Q2772769)

Given a collection of functions \(\{g_m\}_{m\in \mathbb{Z}}\) in \(L^2(\mathbb{R})\) and a parameter \(a>0\), the associated shift-invariant system consists of the functions NEWLINE\[NEWLINE g_{nm}(x)=g_m(x-na),\;m,n\in \mathbb{Z}.NEWLINE\]NEWLINE \(\{g_{nm}\}\) is a frame for \(L^2(\mathbb{R})\) if there exist constants \(A,B>0\) such that NEWLINE\[NEWLINEA\|f\|^2 \leq \sum_{m,n} |\langle f,g_{nm}\rangle|^2\leq B\|f\|^2, \;\forall f\in L^2(\mathbb{R}).NEWLINE\]NEWLINE In this case the associated frame operator on \(L^2(R)\), \ NEWLINE\[NEWLINESf= \sum_{m,n}\langle f,g_{nm}\rangle g_{nm},NEWLINE\]NEWLINE is bounded and invertible, and each \(f\in L^2(\mathbb{R})\) has the representation NEWLINE\[NEWLINEf=\sum_{m,n}\langle f,S^{-1}g_{nm}\rangle g_{nm}.NEWLINE\]NEWLINE A frame might be overcomplete, i.e., representations of \(f\) with other coefficients might exist. A special case of a shift-invariant system occurs when \(g_m(x)=e^{2\pi imbx}g(x)\) for some \(b>0\) and \(g\in L^2(\mathbb{R})\). In this case we obtain (up to an irrelevant factor) the Gabor system \(\{e^{2\pi imbx}g(x-na)\}_{m,n\in \mathbb{Z}}\) .NEWLINENEWLINENEWLINEThe paper is a survey which focusses on characterizations of frames of the shift-invariant type and their consequences for Gabor systems. A key result (due to Ron and Shen) characterizes shift-invariant frames in terms of the biinfinite matrix-valued function whose \(km\)-th entry consists of the sample \(\hat{g}_m(\cdot -k/a)\). Other highlights are characterizations of Gabor frames \(\{e^{2\pi imbx}g(x-na)\}_{m,n\in Z}\) in terms of properties of the Gabor system, where the parameters \(a,b\) are replaced by \(1/b,1/a\). Also, in case \(a,b\) are rational, the frame property is characterized in terms of the Zak transform and the Zibulski-Zeevi matrices. It is well known that if we fix \(a>0\), functions \(g\) decaying sufficiently fast will always generate a frame for sufficiently small choices of \(b\). Several concrete cases of Gabor systems conclude the paper and show how difficult it is to find the exact range for the parameters \(a,b\) leading to a frame for a given function \(g\). Even the ``easiest'' case, where \(g\) is a characteristic function, requires a very detailed and nontrivial analysis. By the way, the graphical illustration of this case is known as the Janssen tie.NEWLINENEWLINEFor the entire collection see [Zbl 0972.00019].