Opuscula Mathematica
Opuscula Math. 35, no. 6 (2015), 957-972

Nontrivial solutions of linear functional equations: methods and examples

Adrienn Varga
Csaba Vincze
Abstract. For a wide class of linear functional equations the solutions are generalized polynomials. The existence of non-trivial monomial terms of the solution strongly depends on the algebraic properties of some related families of parameters. As a continuation of the previous work [A. Varga, Cs. Vincze, G. Kiss, Algebraic methods for the solution of linear functional equations, Acta Math. Hungar.] we are going to present constructive algebraic methods of the solution in some special cases. Explicit examples will be also given.
Keywords: linear functional equations, spectral analysis, field homomorphisms.
Mathematics Subject Classification: 39B22.
Full text (pdf)
References ▼
  1. [1] Z. Daróczy, Notwendige und hinreichende Bedingungen für die Existenz von nichtkonstanten Lösungen linearer Funktionalgleichungen, Acta Sci. Math. Szeged 22 (1961), 31-41.
  2. [2] G. Kiss, A. Varga, Existence of nontrivial solutions of linear functional equations, Aequat. Math. 88 (2014), 151-162.
  3. [3] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Prace Naukowe Uniwersytetu Śląskiego w Katowicach, vol. CDLXXXIX, Państwowe Wydawnictwo Naukowe - Uniwersytet Śląski, Warszawa-Kraków-Katowice, 1985.
  4. [4] M. Laczkovich, G. Kiss, Linear functional equations, differential operators and spectral synthesis, accepted for publication in Aequat. Math., published online 21 June 2014.
  5. [5] M. Laczkovich, G. Kiss, Non-constant solutions of linear functional equations, 49th Int. Symp. on Functional equation, Graz (Austria), June 19-26, 2011, http://www.uni-graz.at/jens.schwaiger/ISFE49/talks/saturday/Laczkovich.pdf
  6. [6] L. Székelyhidi, On a class of linear functional equations, Publ. Math. (Debrecen) 29 (1982), 19-28.
  7. [7] L. Székelyhidi, Convolution type functional equations on topological Abelian goups, World Scientific Publishing Co. Inc. Teaneck, NJ, 1991.
  8. [8] A. Varga, On additive solutions of a linear equation, Acta Math. Hungar. 128 (2010), 15-25.
  9. [9] A. Varga, Cs. Vincze, On Daróczy's problem for additive functions, Publ. Math. Debrecen 75 (2009), 299-310.
  10. [10] A. Varga, Cs. Vincze, On a functional equations containing weighted arithmetic means, International Series of Numerical Mathematics 157 (2009), 305-315.
  11. [11] A. Varga, Cs. Vincze, G. Kiss, Algebraic methods for the solution of linear functional equations, accepted for publication in Acta Math. Hungar.
About the authors ▼
Adrienn Varga
University of Debrecen, H-4010 Debrecen, P.O. Box 12, Hungary
Csaba Vincze
University of Debrecen, H-4010 Debrecen, P.O. Box 12, Hungary
About the article ▼
Communicated by Karol Baron.
Received: 2014-08-28.
Revised: 2014-12-15.
Accepted: 2014-12-16.
Published online: 2015-06-06.
Cite this article as ▼
Adrienn Varga, Csaba Vincze, Nontrivial solutions of linear functional equations: methods and examples, Opuscula Math. 35, no. 6 (2015), 957-972, http://dx.doi.org/10.7494/OpMath.2015.35.6.957.
Download this article's citation as: