aequationes mathematicae

, Volume 53, Issue 1–2, pp 91–107 | Cite as

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).

AMS (1991) subject classification

39B32 

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Copyright information

© Birkhäuser Verlag 1997

Authors and Affiliations

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

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