Abstract
The equation
belongs to the class of linear functional equations. The solutions form a linear space with respect to the usual pointwise operations. According to the classical results of the theory they must be generalized polynomials. New investigations have been started a few years ago. They clarified that the existence of non-trivial solutions depends on the algebraic properties of some related families of parameters. The problem is to find the necessary and sufficient conditions for the existence of non-trivial solutions in terms of these kinds of properties. One of the earliest results is due to Z. Daróczy [1]. It can be considered as the solution of the problem in case of n = 2. We are going to take more steps forward by solving the problem in case of n = 3.
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References
Z. Daróczy, Notwendige und hinreichende Bedingungen für die Existenz von nichtkonstanten Lösungen linearer Funktionalgleichungen, Acta Sci. Math. (Szeged), 22 (1961), 31–41.
Varga A.: On additive solutions of a linear equation. Acta Math. Hungar., 128, 15–25 (2010)
Varga A., Vincze Cs.: On Daróczy’s problem for additive functions. Publ. Math. Debrecen, 75, 299–310 (2009)
A. Varga and Cs. Vincze, On a functional equation containing weighted arithmetic means, International Series of Numerical Mathematics, 157 (2009), 305–315.
Zs. Páles, Extension theorems for functional equations with bisymmetric operators, Aequat. Math., 63 (2002), 266–291.
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).
L. Székelyhidi, On a class of linear functional equations, Publ. Math. Debrecen, 29 (1982), 19–28.
M. Laczkovich and G. Székelyhidi, Harmonic analysis on discrete Abelian groups, Proc. Amer. Math. Soc., 133 (2005), 1581–1586.
G. Kiss and A. Varga, Existence of nontrivial solutions of linear functional equations, Aequat. Math., 88 (2014), 151–162.
M. Laczkovich and G. Kiss, Linear functional equations, differential operators and spectral synthesis, accepted for publication in Aequat. Math.
M. Laczkovich and G. Kiss, Non-constant solutions of linear functional equations, in: 49th Int. Symp. on Functional equation (Graz, Austria), June 19–26 (2011), http://www.uni-graz.at/jens.schwaiger/ISFE49/talks/saturday/Laczkovich.pdf.
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G. Kiss was partially supported by the Hungarian National Foundation for Scientific Research, Grants No. K104178.
A. Varga was supported by the European Union co-financed by the European Social Fund. The work was carried out as a part of the TÁMOP-4.1.1.C-12/1/KONV-2012-0012 project in the framework of the New Hungarian Development Plan.
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.
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Kiss, G., Varga, A. & Vincze, C. Algebraic methods for the solution of linear functional equations. Acta Math. Hungar. 146, 128–141 (2015). https://doi.org/10.1007/s10474-015-0497-6
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DOI: https://doi.org/10.1007/s10474-015-0497-6