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Spherical spectral synthesis

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Abstract

We make an attempt to extend L. Schwartz’s classical result on spectral synthesis to several dimensions. Due to counterexamples of D. I. Gurevich this is impossible for translation invariant varieties. Our idea is to replace translations by proper euclidean motions in higher dimensions. For this purpose we introduce the basic concepts of spectral analysis and synthesis in the non-commutative setting based on Gelfand pairs, where “translation invariance" will be replaced by invariance with respect to a compact group of automorphisms. The role of exponential functions will be played by spherical functions. As an application we obtain the extension of L. Schwartz’s fundamental result.

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Authors and Affiliations

  1. Institute of Mathematics, University of Debrecen, Debrecen, Hungary

    L. Székelyhidi

  2. Department of Mathematics, University of Botswana, Gaborone, Botswana

    L. Székelyhidi

Authors
  1. L. Székelyhidi
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Correspondence to L. Székelyhidi.

Additional information

The research was partly supported by the Hungarian National Foundation for Scientific Research (OTKA), Grant No. K111651.

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Székelyhidi, L. Spherical spectral synthesis. Acta Math. Hungar. 153, 120–142 (2017). https://doi.org/10.1007/s10474-017-0715-5

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  • DOI: https://doi.org/10.1007/s10474-017-0715-5

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