aequationes mathematicae

, Volume 53, Issue 1–2, pp 91–107

Wilson's functional equation on C

• Henrik Stetkaer
Research Papers

Summary

We find the complete set of continuous solutionsf, g of “Wilson's functional equation”Σ n = 0 N − 1 f(x + wny) = Nf(x)g(y), x, y ∈ C, given a primitiveNth rootw of unity.

Disregarding the trivial solutionf = 0 andg any complex function, it is known thatg satisfies a version of d'Alembert's functional equation and so has the formg(z) = gε(z) = N−1 Σ n = 0 N − 1 Eμ(wnz) for someμ ∈ C2. HereE(μ1, μ2)(x + iy) = exp(μ1x + μ2).

For fixedg = gμ the space of solutionsf of Wilson's functional equation can be decomposed into theN isotypic subspaces for the action of Z N on the continuous functions on C. We prove that therth component, wherer ∈ {0, 1, ⋯,N − 1}, of any solution satisfies the signed functional equationΣ n = 0 N − 1 f(x + wny)wnr = Ng(x)f(y), x, y ∈ C. We compute the solution spaces of each of these signed equations: They are 1-dimensional and spanned byz → Σ n = 0 N − 1 wnr Eμ(wnz), except forg = 1 andr ≠ 0 where they are spanned by$$\bar z^r$$ andzN − r. Adding the components we get the solution of Wilson's equation. Analogous results are obtained with the action ofZN on C replaced by that ofSO(2).

The case ofg = 0 in the signed equations is special and solved separately both for Z N andSO(2).

39B32

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References

1. [1]
Aczél, J.,Vorlesungen über Funktionalgleichungen und ihre Anwendungen. Birkhäuser Verlag, Basel—Stuttgart, 1961.Google Scholar
2. [2]
Aczél, J., Haruki, H., McKiernan, M. A. andSakovič, G. N.,General and regular solutions of functional equations characterizing harmonic polynomials. Aequationes Math.1 (1968), 37–53.
3. [3]
Aczél, J. andDhombres, J.,Functional equations in several variables. Cambridge University Press, Cambridge—New York—New Rochelle—Melbourne—Sydney, 1989.Google Scholar
4. [4]
Badora, R.,On a joint generalization of Cauchy's and d'Alembert's functional equations. Aequationes Math.43 (1992), 72–89.
5. [5]
Borel, A.,Représentations de groupes localement compacts. Springer Verlag, Berlin—Heidelberg—New York, 1972.Google Scholar
6. [6]
Burckel, R. B.,An introduction to classical complex analysis. Vol. 1. Birkhäuser Verlag, Basel und Stuttgart, 1979.Google Scholar
7. [7]
Chojnacki, W.,Fonctions cosinus hilbertiennes bornées dans les groupes commutatifs localement compacts. Compositio Math.57 (1986), 15–60.Google Scholar
8. [8]
Chojnacki, W.,On some functional equation generalizing Cauchy's and d'Alembert's functional equations. Colloq. Math.55 (1988), 169–178.Google Scholar
9. [9]
Förg-Rob, W. andSchwaiger, J.,A generalization of the cosine equation. Grazer Math. Ber.315 (1991), 25–34.Google Scholar
10. [10]
Förg-Rob, W. andSchwaiger, J.,On a generalization of the cosine equation to n summands. Grazer Math. Ber.316 (1992), 219–226.Google Scholar
11. [11]
Förg-Rob, W. andSchwaiger, J.,On the stability of some functional equations for generalized hyperbolic functions and for the generalized cosine equation. Resultate Math.26 (1994), 274–280.Google Scholar
12. [12]
Helgason, S.,Groups and geometric analysis. Academic Press, Orlando—San Diego—San Francisco—New York—London—Toronto—Montreal—Sydney—Tokyo—São Paulo, 1984.Google Scholar
13. [13]
Hewitt, E. andRoss, K. A.,Abstract harmonic analysis. Volume II. Springer Verlag, Berlin—Heidelberg—New York, 1970.Google Scholar
14. [14]
Hörmander, L.,The analysis of linear partial differential operators I. Springer Verlag, Berlin—Heidelberg—New York—Tokyo, 1983.Google Scholar
15. [15]
Kannappan, Pl.,The functional equation f(xy) + f(xy −1) = 2f(x)f(y) for groups. Proc. Amer. Math. Soc.19 (1968), 69–74.Google Scholar
16. [16]
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., Teaneck, NJ, 1989.Google Scholar
17. [17]
Schwaiger, J.,On generalized hyperbolic functions and their characterization by functional equations. Aequationes Math.43 (1992), 198–210.
18. [18]
Sinopoulos, P.,Generalized sine equations, I. Aequationes Math.48 (1994), 171–193.
19. [19]
Stetkaer, H.,D'Alembert's equation and spherical functions. Aequationes Math.48 (1994), 220–227.
20. [20]
Stetkaer, H.,Wilson's functional equations on groups. Aequationes Math.49 (1995), 252–275.
21. [21]
Stetkaer, H.,On a signed cosine equation of N summands. [Preprint Series, No. 1994/1]. Matematisk Institut, Aarhus. To appear in Aequationes Math.Google Scholar
22. [22]
Stetkaer, H.,Functional equations and spherical functions. [Preprint Series, No. 1994/18]. Matematisk Institut, Aarhus University, Aarhus, 1994.Google Scholar
23. [23]
Wilson, W. H.,On certain related functional equations. Bull. Amer. Math. Soc.26 (1919–20), 300–312.Google Scholar