Laws of Large Numbers, Spectral Translates and Sampling over LCA Groups
Kluv´anek extended the Whittaker-Kotel’nikov-Shannon theorem to the abstract harmonic analysis setting over a LCA group G. In this context, the classical condition for f ∈ L 2 (R) to be band limited is replaced by fb having its support essentially contained in a transversal set of a compact quotient...
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| Formato: | Artículo |
| Lenguaje: | Inglés |
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Universidad de Buenos Aires
2025
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| Acceso en línea: | https://repositorio.uca.edu.ar/handle/123456789/20837 |
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| Sumario: | Kluv´anek extended the Whittaker-Kotel’nikov-Shannon theorem to the abstract harmonic analysis setting over a LCA group G. In this context, the classical condition for f ∈ L 2 (R) to be band limited is replaced by fb having its support essentially contained in a transversal set of a compact quotient group. This condition was later shown to be necessary in general. Moreover, the classical interpolation formula is also equivalent to a Plancherel like isometric formula involving the L 2 (G) norm of f and the norm of the sequence of its samples over a subgroup H. Here, recalling some Laws of Large Numbers, we will prove an equivalent result for the support of the spectral measure µX of a Gaussian stationary random process X, indexed over a LCA group G. The conditions are formulated in terms of an almost sure isometric formula involving the sample variances of X, and its samples over a subgroup H respectively. |
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