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- W1567716289 abstract "We present results concerning generalized translation invariant (GTI) systems on a second countable locally compact abelian group G. These are systems with a family of generators {g <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>j, P</sub> } <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>jεJ, pεPJ</sub> ⊂ L <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>2</sup> (G), where J is a countable index set, and P <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>j</sub> , j ε J are certain measure spaces. Furthermore, for each j we let Γ <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>j</sub> , be a closed subgroup of G such that G/Γ <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>j</sub> is compact. A GTI system is then the collection of functions U <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>jεJ</sub> {g <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>j, p</sub> (· - γ} <sub xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>γεΓj, pεPj</sub> . Many well known systems, such as wavelet, shearlet and Gabor systems, both the discrete and continuous types, are GTI systems. We characterize when such systems form tight frames, and when two GTI Bessel systems form dual frames for L <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>2</sup> (G). In particular, this offers a unified approach to the theory of discrete and continuous frames and, e.g., yields well known results for discrete and continuous Gabor and wavelet systems." @default.
- W1567716289 created "2016-06-24" @default.
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- W1567716289 date "2015-05-01" @default.
- W1567716289 modified "2023-10-18" @default.
- W1567716289 title "A characterization of tight and dual generalized translation invariant frames" @default.
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- W1567716289 doi "https://doi.org/10.1109/sampta.2015.7148858" @default.
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