Quantum harmonic analysis on lattices and Gabor multipliers (Q785661)

The paper is in a scope of the time-frequency analysis and it is devoted to certain discrete aspects of the theory. If \(\Lambda=A\mathbb Z^{2d}\) for \(A\in GL(2d,\mathbb R)\) and \(m=\{m(\lambda)\}_{\lambda\in\Lambda}\) is a sequence of complex numbers then the Gabor multiplier \(G_m^{\varphi,\xi}: L^2(\mathbb R^d)\rightarrow L^2(\mathbb R^d) \) is defined by the formula \[ G_m^{\varphi,\xi}(\psi)=\sum\limits_{\lambda\in\Lambda}m(\lambda)V_{\varphi}\psi(\lambda)\pi(\lambda)\xi \] where \(\varphi,\xi\in L^2(\mathbb R^d), V_{\varphi}\) is the short-time Fourier transform \[ V_{\varphi}\psi(z)=(\psi,\pi(z)\varphi)_{L^2(\mathbb R^d)}, \] \(\pi(z): L^2(\mathbb R^d)\rightarrow L^2(\mathbb R^d) \) is the time-frequency shift given by \(\pi(z)\varphi(t)=e^{2\pi i\omega\cdot t}\varphi(t-x)\) for \(z=(x,\omega)\). As the author says the main goal is ``to develop a version of quantum harmonic analysis for discrete lattices to provide a similar conceptual framework for Gabor multipliers''. The author introduces a convolutions for sequences and operators from trace class, studies their properties using different technique like Weyl transform, the symplectic Fourier transform etc. Particularly, the Gabor multiplier can be written as a certain convolution. Many results from classical harmonic analysis in \(\mathbb R^d\) are reinterpreted from discrete point of view.