Advertisement

Monatshefte für Mathematik

, Volume 175, Issue 4, pp 639–643 | Cite as

On Fréchet’s functional equation

  • László SzékelyhidiEmail author
Article

Abstract

Fréchet’s functional equation \(\Delta _{y_1,y_2,\dots ,y_{n+1}}f=0\) plays a key role in the theory of polynomial functions. A basic theorem of Djokovič shows that under general conditions the functional equation \(\Delta _y^{n+1}f=0\) is equivalent to Fréchet’s equation. Here we give a short alternative proof for this result using spectral synthesis.

Keywords

Fréchet’s functional equation Spectral synthesis 

Mathematics Subject Classification (2000)

39A70 39A99 

References

  1. 1.
    Djokovič, D.Z.: A representation theorem for \({(X_1-1)(X_2-1){\ldots }(X_n-1)}\) and its applications. Ann. Polon. Math. 22, 189–198 (1969/1970)Google Scholar
  2. 2.
    Fréchet, M.: Une définition fonctionelle des polyn\(\hat{o}\)mes. Nouv. Ann. 49, 145–162 (1909)Google Scholar
  3. 3.
    Lefranc, M.: Analyse spectrale sur \(Z_{n}\). C. R. Acad. Sci. Paris 246, 1951–1953 (1958)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Mazur, S., Orlicz, W.: Grundlegende Eigenschaften der polynomischen Operationen I. Studia Math. 5, 50–68 (1934)zbMATHGoogle Scholar
  5. 5.
    Mazur, S., Orlicz, W.: Grundlegende Eigenschaften der polynomischen Operationen II. Studia Math. 5, 179–189 (1934)zbMATHGoogle Scholar
  6. 6.
    Székelyhidi, L.: Convolution type functional equations on topological abelian groups. World Scientific Publishing Co., Inc., Teaneck (1991)CrossRefzbMATHGoogle Scholar
  7. 7.
    Székelyhidi, L.: The octahedron and cube functional equations revisited. Publ. Math. Debrecen 61(3–4), 241–252 (2002)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Székelyhidi, L.: Difference equations via spectral synthesis. Ann. Univ. Sci. Budapest. Sect. Comput. 24, 3–14 (2004)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Székelyhidi, L.: Discrete spectral synthesis and its applications, Springer Monographs in Mathematics. Springer, Dordrecht (2006)Google Scholar
  10. 10.
    van der Lijn, G.: Les polynomes abstraits I. Bull. Sci. Math. 64, 55–80 (1940)MathSciNetGoogle Scholar
  11. 11.
    van der Lijn, G.: Les polynomes abstraits II. Bull. Sci. Math. (2) 64, 128–144 (1940)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Wien 2013

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of DebrecenDebrecenHungary
  2. 2.Department of MathematicsUniversity of BotswanaGaboroneBotswana

Personalised recommendations