The aim of the paper is to investigate the solutions of special inhomogeneous linear functional equations using spectral analysis in a translation invariant closed linear subspace of additive/multiadditive functions containing the restrictions of the solutions to finitely generated fields. The application of spectral analysis in some related varieties is a new and important trend in the theory of functional equations; especially they have successful applications in the case of homogeneous linear functional equations. The foundations of the theory can be found in Kiss and Varga (Aequat Math 88(1):151–162, 2014) and Kiss and Laczkovich (Aequat Math 89(2):301–328, 2015). We are going to adopt the main theoretical tools to solve some inhomogeneous problems due to Koclȩga-Kulpa and Szostok (Ann Math Sylesianae 22:27–40, 2008), see also Koclȩga-Kulpa and Szostok (Georgian Math J 16:725–736, 2009; Acta Math Hung 130(4):340–348, 2011). They are motivated by quadrature rules of approximate integration.
Linear functional equations Spectral analysis Spectral synthesis
Mathematics Subject Classification
Primary 43A45 43A70 Secondary 13F20
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Vincze, Cs: Algebraic dependency of roots of multivariate polynomials and its applications to linear functional equations. Period. Math. Hung. 74(1), 112–117 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
Vincze, Cs, Varga, A.: On a sufficient and necessary condition for a multivariate polynomial to have algebraic dependent roots–an elementary proof. Acta Math. Acad. Paedagog. Nyházi 33, 1–13 (2017)MathSciNetGoogle Scholar