On the Product of Localization Operators

Abstract. We provide examples of the product of two localization operators. As a special case, we study the composition of Gabor multipliers. The re-sults highlight the instability of this product and underline the necessity of expressing it in terms of asymptotic expansions. We study the problem of calculating or estimating the product of two local-ization operators. The motivation comes either from signal analysis or pseudodiffe-rential operator theory. On the one hand, in signal analysis the problem of finding a filter that has the same effect as two filters arranged in series amounts to the computation of the product of two localization operators, see [8, 9] and references therein. On the other side, composition of pseudodifferential operators by means of symbolic calculus gives rise to asymptotic expansions, mainly employed in PDEs (see e.g., [14, 16]). Outcomes are regularity properties of partial differential oper-ators and the construction of an approximate inverse (so-called parametrix). Since localization operators are a sub-class of pseudodifferential operators, looking for asymptotic expansions of the localization operator product appears to be natural as well. Applications can be found in the framework of PDEs and Quantum Mechanics [1, 6, 7, 15]. In this paper, we survey the known approaches to this problem and provide concrete examples of the composition of localization operators. Indeed, very few cases allow the product to be written as a localization operator as well, conse-quently the class of localization operators is not closed under composition. Thus the product is unstable with respect to composition. This instability highlights the importance of a symbolic calculus for localization operators [6]. We present localization operators using language and tools from time-frequency analysis. First, the definition of the short-time Fourier transform is required. Given a function f on Rd and a point (x, ω) of the phase space R2d, the operators of translation and modulation are defined to be (1) Txf(t) = f(t − x) and Mωf(t) = e2piiωtf(t)