Density of Gabor frames
A Gabor system is a system of time-frequency shifts \(S(g,\Lambda)=\{ e^{2\pi i bx}g(x-a){\}}_{(a,b)\in\Lambda}\) of a function \(g\in L^2(R)\) and a set of time-frequency shifts \(\Lambda\subset R^{2d}\). The authors obtain necessary conditions for a finite union \(S=\bigcup_{k=1}^rS(g_k,\Lambda_k)\) to be frame or Riesz basis, in terms of Beurling densities of \(\Lambda=\bigcup_{k=1}^r\Lambda_k\). They show (Theorem 1.1 and Corollary 3.7): (i) if \(S\) is a frame then \(1\leq D^{-}(\Lambda)\leq D^{+}(\Lambda)<\infty\); (ii) if \(S\) is a Riesz basis for \(L^2(R^d)\), then \(D^{+}(\Lambda)=D^{-}(\Lambda)=1\). In particular, it is shown that no collection \(\bigcup_{k=1}^r\{g_k(x-a){\}}_{a\in\Gamma_k}\) of pure translates can form a frame for \(L^2(R^d)\) (Theorem 1.2).
scientific article; zbMATH DE number 1384042
scientific article