Incompleteness of sparse coherent states
The abstract of this paper reads: ``A phase space shift of a square integrable function \(\varphi \in L^2 (\mathbb{R}^n)\) is defined by \[ \rho (p,q) \varphi (x) : = e^{2 \pi iqx} \varphi (x + p). \] Let \(\Lambda \subset \mathbb{R}^{2n}\) be a discrete subset and \[ {\mathcal S}_{\Lambda, \varphi} = \bigl\{ \rho (p,q) \varphi : (p,q) \in \Lambda \bigr\}. \] The issue of when an arbitrary function \(f \in L^2 (\mathbb{R}^n)\) can be expanded in terms of a particular set of coherent states, \({\mathcal S}_{\Lambda, \varphi}\) arises in various subjects such as quantum mechanics and signal processing. Special cases lead to the intuition that the possibility of such a general expansion is highly dependent on the density of the set \(\Lambda\). In this paper, we give rigorous results about this relationship. Our approach provides an elementary proof of a result of Rieffel as well as an extension of work of Landau. Our final result states that if \({\mathcal S}_{\Lambda, \varphi}\) is a Riesz basis for \(L^2\), then the asymptotic density of \(\Lambda\) is exactly one.'' In addition, it should be mentioned that the main auxiliary result of this paper is a simple and powerful ``comparison theorem''. This result is of a quite general nature, and it is intrinsically interesting. In a recent paper [\textit{K. Gröchening} and \textit{H. Razafinjatovo}, ``On Landau's necessary density for sampling and interpolation of band-limited functions'', J. Lond. Math. Soc. (to appear)], it is shown how this comparison principle yields certain celebrated results of Henry Landau. Roughly speaking, this principle, which is proved by elementary operator theory, says that a set of points in phase space corresponding to a frame of coherent states is uniformly at least as thick as one corresponding to a Riesz basis of coherent states. Other variants can be proved, e.g., one can establish a similar connection between Riesz sequences and bases, and ``phase space'' and ``coherent states'' can, depending on the context, be replaced by other reasonable index sets and families of vectors.
scientific article; zbMATH DE number 772197