Quantization of TF lattice-invariant operators on elementary LCA groups

Abstract

Elementary locally compact abelian groups $\mathcal{G}$ are a natural setup for an abstract view on time-frequency (TF) analysis. The function space Gelfand triple (S ₀, L ², S′₀)$(\mathcal{G})$ is adapted to the sampling and periodization procedures on the abstract TF-plane ${\cal G} \times \widehat {\cal G}$ and it allows the definition of a generalized Kohn-Nirenberg correspondence for a “harmonic analysis and synthesis” of linear operators. We extend the concept of duality and biorthogonality of Gabor atoms to arbitrary discrete subgroups of ${\cal G} \times \widehat {\cal G}$ with compact quotient. The setting of elementary LCA groups is not only an extension of standard Gabor analysis but admits a unified formulation for continuous-time, discrete-time, periodic, and multidimensional signals including the case of nonseparable lattices and/or nonseparable atoms.