On the characteristic polynomials of linear functional equations
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The solutions of a linear functional equation are typically generalized polynomials. The existence of their non-trivial monomial terms strongly depends on the algebraic properties of some related families of parameters. In extremal cases (the parameters are algebraic numbers or the parameters form an algebraically independent system) we have elegant methods to decide the existence of non-trivial solutions. In this paper we are going to extend and unify the treatment of the existence problem by introducing the characteristic polynomials of a linear functional equation such that the algebraic properties of the roots allows us to conclude the existence of non-trivial solutions.
KeywordsLinear functional equations Spectral analysis Field homomorphisms
Mathematics Subject Classification39B22
Cs. Vincze was partially supported by the European Union and the European Social Fund through the project Supercomputer, the national virtual lab (Grant No.:TÁMOP-4.2.2.C-11/1/KONV-2012-0010). The work is supported by the University of Debrecen’s internal research Project RH/885/2013.
- 2.M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Prace Naukowe Universitetu Śla̧skiego w Katowicach Vol. CDLXXXIX (Państwowe Wydawnictwo Naukowe – Universitet Śla̧ski, Warszawa–Kraków–Katowice, 1985)Google Scholar
- 3.M. Laczkovich, G. Kiss, Linear functional equations, differential operators and spectral synthesis. Aequat. Math. 89(2), 301–328 (2015)Google Scholar
- 4.M. Laczkovich, G. Kiss, Non-constant solutions of linear functional equations, 49th international Symposium on functional equation, Graz (Austria), June 19–26, 2011, http://www.uni-graz.at/jens.schwaiger/ISFE49/talks/saturday/Laczkovich
- 8.A. Varga, Cs. Vincze, On Daróczy’s problem for additive functions. Publ. Math. Debrecen 75(1–2), 299–310 (2009)Google Scholar
- 9.A. Varga, Cs. Vincze, On a functional equations containing weighted arithmetic means. Int. Ser. Numer. Math. 157, 305–315 (2009)Google Scholar
- 10.A. Varga, Cs. Vincze, Nontrivial solutions of linear functional equations: methods and examples. Opusc. Math. 35(6), 957–972 (2015)Google Scholar