Skip to main content
SpringerLink
Log in

Functional equations on abelian groups with involution

  • Research Papers
  • Published:
aequationes mathematicae Aims and scope Submit manuscript

Summary

We produce complete solution formulas of selected functional equations of the formf(x +y) ±f(x + σ (ν)) = Σ 2 I =1 g l (x)h l (y),x, y∈G, where the functionsf,g 1,h 1 to be determined are complex valued functions on an abelian groupG and where σ:G→G is an involution ofG. The special case of σ=−I encompasses classical functional equations like d’Alembert’s, Wilson’s first generalization of it, Jensen’s equation and the quadratic equation. We solve these equations, the equation for symmetric second differences in product form and similar functional equations for a general involution σ.

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. Aczél, J.,The general solution of two functional equations by reduction to functions additive in two variables and with the aid of Hamel bases. Glasnik Mat.-Fiz Astronom. Ser. II Društvo Mat. Fiz. Hrvatske20 (1965), 65–73.

    MATH  Google Scholar 

  2. Aczél, J.,Vorlesungen über Funktionalgleichungen und ihre Anwendungen. Birkhäuser, Basel/Stuttgart 1961.

    MATH  Google Scholar 

  3. Aczél, J., Chung, J. K. andNg, C. T.,Symmetric second differences in product form on groups. Topics in mathematical analysis (pp. 1–22) edited by Th. M. Rassias. Ser. Pure Math.,11, World Scientific Publ. Co., Teaneck, NJ 1989.

    Google Scholar 

  4. Aczél, J. andDhombres, J.,Functional equations in several variables. Cambridge University Press, Cambridge/New York/New Rochelle/Melbourne/Sydney 1989.

    MATH  Google Scholar 

  5. Aczél, J. andVincze, E.,Über eine gemeinsame Verallgemeinerung zweier Funktionalgleichungen von Jensen. Publ. Math. Debrecen10 (1963), 326–344.

    MathSciNet  Google Scholar 

  6. Badora, R.,On a joint generalization of Cauchy’s and d’Alembert’s functional equations. Aequationes Math.43 (1992), 72–89.

    Article  MATH  MathSciNet  Google Scholar 

  7. Baker, J. A.,The stability of the cosine equation. Proc. Amer. Math. Soc.80 (1980), 411–416.

    Article  MATH  MathSciNet  Google Scholar 

  8. Chojnacki, W.,Fonctions cosinus hilbertiennes bornées dans les groupes commutatifs localement compacts. Compositio Math.57 (1986), 15–60.

    MATH  MathSciNet  Google Scholar 

  9. Chojnacki, W.,On some functional equation generalizing Cauchy’s and d’Alembert’s functional equations. Colloq. Math.55 (1988), 169–178.

    MATH  MathSciNet  Google Scholar 

  10. Chung, J. K., Ebanks, B. R., Ng, C. T. andSahoo, P. K.,On a quadratic-trigonometric functional equation and some applications. Trans. Amer. Math. Soc.347 (1995), 1131–1161.

    Article  MATH  MathSciNet  Google Scholar 

  11. Corovei, I.,The functional equation f(xy)+f(xy −1)=2f(x)g(y) for nilpotent groups. Mathematica (Cluj)22 (45) (1980), 33–41.

    MathSciNet  Google Scholar 

  12. Drygas, H.,Quasi-inner products and their applications. In:Advances in Multivariate Statistical Analysis (ed. A. K. Gupta), D. Reidel Publishing Co. 1987, 13–30.

  13. Dunford, N. andSchwartz, J. T.,Linear Operators. Part I: General Theory. Interscience Publishers, Inc., New York, 1958.

    Google Scholar 

  14. Ebanks, B. R., Kannappan, Pl. andSahoo, P. K.,A common generalization of functional equations characterizing normed and quasi-inner-product spaces. Canad. Math. Bull.35 (1992), 321–327.

    MathSciNet  Google Scholar 

  15. Förg-Rob, W. andSchwaiger, J.,On a generalization of the cosine equation to n summands. Grazer Math. Ber.316 (1992), 219–226.

    MATH  Google Scholar 

  16. Friedman, D.,The functional equation f(x+y)=g(x)+h(y). Amer. Math. Monthly69 (1962), 769–772.

    Article  MATH  MathSciNet  Google Scholar 

  17. Gajda, Z.,On functional equations associated with characters of unitary representations of groups. Aequationes Math.44 (1992), 109–121.

    Article  MATH  MathSciNet  Google Scholar 

  18. Hewitt, E. andRoss, K. A.,Abstract Harmonic Analysis.Volume II. Springer-Verlag, Berlin-Heidelberg-New York 1970.

    MATH  Google Scholar 

  19. Kaczmarz, S.,Sur l’équation fonctionelle f(x)+f(x+y)=ϕ(y)f(x+(y/2)). Fund. Math.6 (1924), 122–129.

    Google Scholar 

  20. Kannappan, Pl.,The functional equation f(xy)+f(xy −1)=2f(x)f(y)for groups. Proc. Amer. Math. Soc.19 (1968), 69–74.

    Article  MATH  MathSciNet  Google Scholar 

  21. Kannappan, Pl.,A note on cosine functional equation for groups. Mat. Vesnik8 (1971), 317–319.

    MathSciNet  Google Scholar 

  22. Kannappan, Pl.,Cauchy equations and some of their applications. Topics in mathematical analysis (pp. 518–538) edited by Th. M. Rassias. Ser. Pure Math., 11, World Scientific Publ. Co., Teaneck, NJ 1989.

    Google Scholar 

  23. Kisyński, J.,On operator-valued solutions of d’Alembert’s functional equation, I. Colloq. Math.23 (1971), 107–114.

    MathSciNet  MATH  Google Scholar 

  24. Lashkarizadeh Bami, M.,Functional equations and *-representations on topological semigroups. Manuscripta Math.82 (1994), 261–276.

    Article  MATH  MathSciNet  Google Scholar 

  25. Ljubenova, E. T.,On D’Alembert’s functional equation on an Abelian group. Aequationes Math.22 (1981), 54–55.

    Article  MathSciNet  Google Scholar 

  26. O’Connor, Thomas A.,A solution of D’Alembert’s functional equation on a locally compact Abelian group. Aequationes Math.15 (1977), 235–238.

    Article  MATH  MathSciNet  Google Scholar 

  27. Sinopoulos, P.,Generalized sine equations, I. Aequationes Math.48 (1994), 171–193.

    Article  MATH  MathSciNet  Google Scholar 

  28. Stetkær, H.,D’Alembert’s equation and spherical functions. Aequationes Math.48 (1994), 220–227.

    Article  MathSciNet  MATH  Google Scholar 

  29. Stetkær, H.,Wilson’s functional equations on groups. Aequationes Math.49 (1995), 252–275.

    Article  MathSciNet  MATH  Google Scholar 

  30. Stetkær, H.,Functional equations and spherical functions. Preprint Series 1994 No. 18, Matematisk Institut, Aarhus University, Denmark, pp. 1–28.

  31. Stetkær, H.,Wilson’s functional equation on C. Aequationes Math.53 (1997), 91–107.

    Article  MathSciNet  MATH  Google Scholar 

  32. Székelyhidi, L.,Functional equations on abelian groups. Acta Math. Acad. Sci. Hungar.37 (1981), 235–243.

    Article  MATH  MathSciNet  Google Scholar 

  33. Vincze, E.,Eine allgemeinere Methode in der Theorie der Funktionalgleichungen, III. Publ. Math. Debrecen10 (1963), 191–202.

    Google Scholar 

  34. Walsh, J. L.,A mean value theorem for polynomials and harmonic polynomials. Bull. Amer. Math. Soc.42 (1936), 923–930.

    MATH  MathSciNet  Google Scholar 

  35. Wilson, W. H.,On certain related functional equations. Bull. Amer. Math. Soc.26 (1919), 300–312.

    Google Scholar 

  36. Wilson, W. H.,Two general functional equations. Bull. Amer. Math. Soc.31 (1925), 330–334.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Aarhus University, Ny Munkegade, DK-8000, Aarhus C, Denmark

    H. Stetkær

Authors
  1. H. Stetkær
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. Stetkær.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Stetkær, H. Functional equations on abelian groups with involution. Aequ. Math. 54, 144–172 (1997). https://doi.org/10.1007/BF02755452

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02755452

AMS (1991) subject classification

Search

Navigation