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On spectral synthesis in varieties containing the solutions of inhomogeneous linear functional equations

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Abstract

As a continuation of our previous work [2] the aim of the recent paper is to investigate the solutions of special inhomogeneous linear functional equations by using spectral synthesis in translation invariant closed linear subspaces of additive/multiadditive functions containing the restrictions of the solutions to finitely generated fields. The idea is based on the fundamental work of [5]. Using spectral analysis in some related varieties we can prove the existence of special solutions (automorphisms) of the functional equation but spectral synthesis allows us to describe the entire space of solutions on a large class of finitely generated fields. It is spanned by the so-called exponential monomials which can be given in terms of automorphisms of \({\mathbb C}\) and differential operators. We apply the general theory to some inhomogeneous problems motivated by quadrature rules of approximate integration [8], see also [7, 9].

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Authors and Affiliations

  1. Mathematics Research Unit, FSTC, University of Luxembourg, 6, rue Coudenhove-Kalergi, 1359, Luxembourg, Luxembourg

    Gergely Kiss

  2. Department of Geometry, Institute of Mathematics Faculty of Science and Technology, University of Debrecen, P.O. Box 12, Debrecen, 4010, Hungary

    Csaba Vincze

Authors
  1. Gergely Kiss
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  2. Csaba Vincze
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Correspondence to Gergely Kiss.

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The authors declare that they have no conflict of interests.

Additional information

G. Kiss is supported by the Internal Research Project R-STR-1041-00-Z of the University of Luxembourgh and by the Hungarian National Foundation for Scientific Research, Grant No. K104178. Cs. Vincze is supported by the University of Debrecen’s internal research project RH/885/2013.

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Kiss, G., Vincze, C. On spectral synthesis in varieties containing the solutions of inhomogeneous linear functional equations. Aequat. Math. 91, 691–723 (2017). https://doi.org/10.1007/s00010-017-0482-y

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