Harmonic Analysis and Functional Equations

  • Henrik Stetkær
Conference paper
Part of the Trends in Mathematics book series (TM)


Functional equations occur in many parts of mathematics, also in harmonic analysis. As an example we mention that the complex exponential function 7 :x ? exp (?x) for any ? ∈ R is a solution of Cauchy’s functional equation
$$\gamma \left( {x + y} \right) = \gamma \left( x \right)\gamma \left( y \right),\;x,y \in R$$


Functional Equation Haar Measure Spherical Function Continuous Homomorphism Aequationes Math 
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  1. [1]
    Aczél, J., Chung, J. K. and Ng, 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
  2. [2]
    Aczél, J. and Dhombres, J.:Functional equations in several variables . Cambridge University Press Cambridge/New York/New Rochelle/ Melbourne/Sydney 1989.MATHCrossRefGoogle Scholar
  3. [3]
    Aczél, J., Haruki, H., McKiernan, M. A. and Sakovic, G. N.:General and Regular Solutions of Functional Equations Characterizing Harmonic Polynomials. Aequationes Math. 1 (1968), 37–53.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    d’Alembert, J.,Recherches sur la courbe que forme une corde tendue mise en vibration, I-II. Hist. Acad. Berlin (1747) 214–249.Google Scholar
  5. [5]
    Badora, R., Ona joint generalization of Cauchy’s and d’Alembert’s functional equations. Aequationes Math. 43 (1992), 72–89.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Baker, J. A.,The stability of the cosine equation. Proc. Amer. Math. Soc. 80 (1980), 411–416.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Chojnacki, W.,Fonctions cosinus hilbertiennes bornées dans les groupes commutatifs localement compacts. Compositio Math. 57 (1986), 15–60.MathSciNetMATHGoogle Scholar
  8. [8]
    Chojnacki, W.,On some functional equations generalizing Cauchy’s and d’Alembert’s functional equations. Colloq. Math. 55 (1988), 169–178.MathSciNetMATHGoogle Scholar
  9. [9]
    Chung, J. K. (Zhong Jukang), Ebanks, B. R., Ng, C. T., Sahoo, P. K. and Zeng, W. B.,On generalized rectangular and rhombic functional equations. Publ. Math. (Debrecen) 47 (1995), 249–270.MathSciNetMATHGoogle Scholar
  10. [10]
    Chung, J. K., Kannappan, Pl. and Ng, C. T.,On two trigonometric functional equations. Mathematics Reports Toyama University 11 (1988), 153–165.MathSciNetMATHGoogle Scholar
  11. [11]
    Helgason, S.:“Groups and Geometric Analysis.” Academic Press, Inc., Orlando — San Diego — San Francisco — New York — London Toronto — Montreal — Sydney — Tokyo — Sao Paulo 1984.MATHGoogle Scholar
  12. [12]
    Hewitt, E. and Ross, K. A.: Abstract Harmonic Analysis IF. Springer-Verlag. Berlin-Heidelberg-New York 1970.Google Scholar
  13. [13]
    Kannappan, Pl.,The functional equation f(xy)+f(xy 1 ) = 2f(x)f(y) for groups .Proc. Amer. Math. Soc. 19 (1968), 69–74.MathSciNetMATHGoogle Scholar
  14. [14]
    Poulsen, T. and Stetkaer, H.,Functional equations on abelian groups with involution. Preprint Series 1995 No 16, Matematisk Institut, Aarhus University, Denmark, pp. 1–23.Google Scholar
  15. [15]
    Stetkaer, H.,D’Alembert’s equation and spherical functions. Aequationes Math. 48 (1994), 220–227.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Stetkaer, H.,Wilson’s Functional Equations on Groups. Aequationes Math. 49 (1995), 252–275.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    Stetkaer, H.,Functional Equations and Spherical Functions. Preprint Series 1994 No 18, Matematisk Institut, Aarhus University, Denmark, pp. 1–28.Google Scholar
  18. [18]
    Stetkaer, H.,Wilson’s functional equation on C. Preprint Series 1995 No 1, Matematisk Institut, Aarhus University, Denmark, pp. 1–15. Accepted for publication by Aequationes Math.Google Scholar
  19. [19]
    Stetkaer, H.,Functional equations on abelian groups with involution. Accepted for publication by Aequationes Math.Google Scholar
  20. [20]
    Wilson, W. H.,On certain related functional equations. Bull. Amer. Math. Soc. 26 (1919), 300–312.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Henrik Stetkær
    • 1
  1. 1.Department of MathematicsAarhus UniversityAarhus CDenmark

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