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.
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© 1998 Springer Science+Business Media New York
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Feichtinger, H.G., Kozek, W. (1998). Quantization of TF lattice-invariant operators on elementary LCA groups. In: Feichtinger, H.G., Strohmer, T. (eds) Gabor Analysis and Algorithms. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-2016-9_8
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DOI: https://doi.org/10.1007/978-1-4612-2016-9_8
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-7382-0
Online ISBN: 978-1-4612-2016-9
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