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Extra Invariance of Group Actions

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Abstract

Given discrete groups \(\Gamma \subset \Delta \) we characterize \((\Gamma ,\sigma )\)-invariant spaces that are also invariant under \(\Delta \). This will be done in terms of subspaces that we define using an appropriate Zak transform and a particular partition of the underlying group. On the way, we obtain a new characterization of principal \((\Gamma ,\sigma )\)-invariant spaces in terms of the Zak transform of its generator. This result is in the spirit of the well-known characterization of shift-invariant spaces in terms of the Fourier transform. As a consequence of our results, we give a solution for the problem of finding the \((\Gamma ,\sigma )\)-invariant space nearest—in the sense of least squares—to a given set of data.

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Authors and Affiliations

  1. Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, 1428, Buenos Aires, Argentina

    C. Cabrelli, C. A. Mosquera & V. Paternostro

  2. IMAS-CONICET, Consejo Nacional de Investigaciones Científicas y Técnicas, Buenos Aires, Argentina

    C. Cabrelli, C. A. Mosquera & V. Paternostro

Authors
  1. C. Cabrelli
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  2. C. A. Mosquera
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  3. V. Paternostro
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Correspondence to V. Paternostro.

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The research of the authors is partially supported by Grants: UBACyT 20020170100430BA, PICT 2014-1480 (ANPCyT) and CONICET PIP 11220150100355. In particular VP is also supported by UBACyT 20020170200057BA and PICT-2016-2616 (Joven)

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Cabrelli, C., Mosquera, C.A. & Paternostro, V. Extra Invariance of Group Actions. J Geom Anal 31, 11878–11898 (2021). https://doi.org/10.1007/s12220-021-00704-2

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  • DOI: https://doi.org/10.1007/s12220-021-00704-2

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