Skip to main content
SpringerLink
Log in

Existence of nontrivial solutions of linear functional equations

  • Published:
Aequationes mathematicae Aims and scope Submit manuscript

Abstract

In this paper we study the functional equation

$$\sum_{i=1}^n a_i f(b_i x+c_i h)=0 \quad (x, h \in \mathbb{C})$$

where a i , b i , c i are fixed complex numbers and \({f \colon \mathbb{C} \to \mathbb{C}}\) is the unknown function. We show, that if there is i such that \({b_i / c_i \neq b_j /c_j}\) holds for any \({1 \leq j \leq n,\ j \neq i}\), the functional equation has a nonconstant solution if and only if there are field automorphisms \({\phi_1, \ldots, \phi_k}\) of \({\mathbb{C}}\) such that \({\phi_1 \cdots \phi_k}\) is a solution of the equation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Baker, J.A.: A general functional equation and its stability. Proc. Am. Math. Soc. 133 (2004), no. 6, 1657–1664 (2005)

    Google Scholar 

  2. Baron K., Jarczyk W.: Recent results on functional equations in a single variable, perspectives and open problems. Aequationes Math. 61, 1–48 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Kuczma, M.: An introduction to the theory of functional equations and inequalities: Cauchy’s equation and Jensen’s inequality. Polish Scientific Publishers and Silesian University, Warsawa-Kraków-Katowice (1985)

  4. Laczkovich M.: Polynomial mappings on Abelian groups. Aequationes Math. 68, 177–199 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Laczkovich M., Székelyhidi G.: Harmonic analysis on dicrete abelien groups. Proc. Am. Math. Soc. 133(6), 1581–1586 (2004)

    Article  Google Scholar 

  6. Mazur, S., Orlicz, W.: Grundlegende Eigenschaften der polynomischen Operationen. Studia Math. 5, 50-68, 179–189 (1934)

    Google Scholar 

  7. Stević S.: Behaviour of solutions of some linear functional equations at infinity. Appl. Math. Comput. 219, 6134–6141 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  8. Székelyhidi L.: Note on exponential polynomials. Pac. J. Math. 103, 583–587 (1982)

    Article  MATH  Google Scholar 

  9. Székelyhidi L.: On a class of linear functional equations. Publ. Math. (Debrecen) 29, 19–28 (1982)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/C, 1117, Budapest, Hungary

    Gergely Kiss

  2. Faculty of Engineering, University of Debrecen, Pf. 12, 4010, Debrecen, Hungary

    Adrienn Varga

Authors
  1. Gergely Kiss
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Adrienn Varga
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Adrienn Varga.

Additional information

G. Kiss was supported by Hungarian Scientific Foundation grant no. 72655.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kiss, G., Varga, A. Existence of nontrivial solutions of linear functional equations. Aequat. Math. 88, 151–162 (2014). https://doi.org/10.1007/s00010-013-0212-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00010-013-0212-z

Mathematics Subject Classification (1991)

Keywords

Search

Navigation