An integral functional equation on groups under two measures.

Authors

  • B. Fadli IBN Tofail University.
  • D. Zeglami Moulay Ismail University.
  • S. Kabbaj IBN Tofail University.

Keywords:

Functional equation, Van Vleck, Kannappan, Involutive automorphism, Group character

Abstract

Let G be a locally compact Hausdorff group, let σ be a continuous involutive automorphism on G, and let µ, ν be regular, compactly supported, complex-valued Borel measures on G. We find the continuous solutions f : G → C of the functional equation

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in terms of continuous characters of G. This equation provides a common generalization of many functional equations (d’Alembert’s, Cauchy’s, Gajda’s, Kannappan’s, Stetkær’s, Van Vleck’s equations...). So, a large class of functional equations will be solved.

 

 

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Author Biographies

  • B. Fadli, IBN Tofail University.

    Department of Mathematics.
    Faculty of Sciences.

  • D. Zeglami, Moulay Ismail University.

    E. N. S. A. M.,

    Department of Mathematics.

  • S. Kabbaj, IBN Tofail University.

    Department of Mathematics.

    Faculty of Sciences.

References

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Fadli, B., Zeglami, D., Kabbaj, S.: A joint generalization of Van Vleck’s and Kannappan’s equations on groups. Adv. Pure Appl. Math. 6(3), pp. 179-188, (2015).

Gajda, Z.: A generalization of d’Alembert’s functional equation, Funkcial. Ekvac. 33 (1), pp. 69-77, (1990).

Kannappan, PL.: A functional equation for the cosine. Can. Math. Bull. 11, pp. 495-498, (1968).

Kannappan, PL.: Functional equations and inequalities with applications. Springer, New York, (2009).

Perkins, A.M., Sahoo, P.K.: On two functional equations with involution on groups related to sine and cosine functions. Aequationes Math. 89 (5), pp. 1251-1263, (2015).

Stetkær, H.: Functional equations on groups. World Scientific, Publishing Co, Singapore, (2013).

Stetkær, H.: Van Vleck’s functional equation for the sine. Aequationes Math. 90 (1), pp. 25-34, (2016).

Stetkær, H.: Kannappan’s functional equation on semigroups with involution. Semigroup Forum. 94 (1), pp. 17-30, (2017).

Van Vleck, E.B.: A functional equation for the sine. Ann. Math. 7, pp. 161-165, (1910).

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Published

2018-09-25

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Section

Artículos

How to Cite

[1]
“An integral functional equation on groups under two measures”., Proyecciones (Antofagasta, On line), vol. 37, no. 3, pp. 565–581, Sep. 2018, Accessed: Sep. 19, 2024. [Online]. Available: https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/3172