Skip to main content
SpringerLink
Log in

Generalized sine equations, II

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

Summary

In this paper we solve the equations

$$\begin{gathered} f(x + y)f(x - y) = \sum\limits_{ij = 0}^2 {\alpha _{ij} f(x)^i f(y)^j ,} \hfill \\ f(x + y)f(x - y) = \sum\limits_{j = 0}^2 {f_j (x)f(y)^j ,} \hfill \\ \end{gathered} $$

which generalize the sine and cosine (d'Alembert) equations. A conditional d'Alembert—Wilson equation and some related unconditional equations in more than two variables are also considered.

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.,Lectures on functional equations and their applications. Academic Press, New York, 1966.

    Google Scholar 

  2. Aczél, J., Chung, J. K. andNg, C. T.,Symmetric second differences in product form on groups. InTopics in mathematical analysis (edited by Th. M. Rassias). World Scientific Publ., Singapore, 1989, pp. 1–22.

    Google Scholar 

  3. Aczél, J. andDhombres, J.,Functional equations in several variables. Cambridge Univ. Press, Cambridge, 1989.

    Google Scholar 

  4. Baker, J. A.,On the functional equation f(x)g(y) = Π n l = 1 h l (a i x + b i y). Aequationes Math.11 (1974), 154–162.

    Google Scholar 

  5. Jordan, P. andvon Neumann, J.,On inner products in linear metric spaces. Ann. Math.36 (1935), 719–723.

    MathSciNet  Google Scholar 

  6. Kannappan, Pl.,The functional equation f(xy) + f(xy −1 = 2f(x)f(y) for groups. Proc. Amer. Math. Soc.29 (1968), 69–74.

    Google Scholar 

  7. Lang, S.,Algebra, Second edition. Addison-Wesley, Reading, MA, 1984.

    Google Scholar 

  8. Perelli, A. andZannier, U.,Su un'equazione funzionale legata agli omomorfismi di un gruppo in un corpo. Boll. Un. Mat. Ital. B (6)5-B (1986), 235–245.

    Google Scholar 

  9. Reich, L. andSchwaiger J.,On polynomials in additive and multiplicative functions. InFunctional equations: history, applications and theory (edited by J. Aczél). Reidel, Dordrecht, 1984, pp. 127–160.

    Google Scholar 

  10. Rukhin, A. L.,The solution of the functional equation of d'Alembert's type for commutative groups. Internat. J. Math. Math. Sci.5 (1982), 315–335.

    Article  Google Scholar 

  11. Sinopoulos, P.,A new generalization of sine and cosine functional equations. Bull. Greek Math. Soc.28 (1987), part B, 43–49.

    Google Scholar 

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

    Article  Google Scholar 

  13. Székelyhidi, L.,Regularity properties of exponential polynomials on groups. Acta Math. Hungar.45 (1985), 21–26.

    Google Scholar 

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

    Google Scholar 

  15. Vincze, E.,Über ein Funktionalgleichungsproblem von I. Olkin und dessen Verallgemeinerungen. Publ. Techn. Univ. Heavy Industry (Miskolc) Ser. D33 (1978), 113–124.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. 18 Vergovitsas Street, GR-11475, Athens, Greece

    Pavlos Sinopoulos

Authors
  1. Pavlos Sinopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sinopoulos, P. Generalized sine equations, II. Aeq. Math. 49, 122–152 (1995). https://doi.org/10.1007/BF01827933

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01827933

AMS (1991) subject classification

Search

Navigation