Laws of Large Numbers, Spectral Translates and Sampling over LCA Groups

• Autores: Medina, Juan Miguel
• Año de publicación: 2022
• Idioma: inglés
• Tipo de recurso: artículo
• Fuente: Workshop in Harmonic Analysis, Sampling Theory, Machine Learning, and Data Science. 2022
• Materia: MATEMATICA; MUESTREO; ANALISIS ARMÓNICO ABSTRACTO
• Institución: Pontificia Universidad Católica Argentina

Fil: Medina, Juan Miguel. Universidad de Buenos Aires, Facultad de Ingeniería, Depto. de Matemática and Instituto Argentino de Matemática; Argentina
Fil: Medina, Juan Miguel. Pontificia Universidad Católica Argentina, Facultad de Ingeniería y Cs. Agrarias; Argentina
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.