Abstract
We are looking for local analytic respectively formal solutions of the generalized Dhombres functional equation \({f(zf(z))=\varphi(f(z))}\) in the complex domain. First we give two proofs of the existence theorem about solutions f with f(0) = w 0 and \({w_0 \in \mathbb{C}^\star {\setminus}\mathbb{E}}\) where \({\mathbb{E}}\) denotes the group of complex roots of 1. Afterwards we represent solutions f by means of infinite products where we use on the one hand the canonical convergence of complex analysis, on the other hand we show how solutions converge with respect to the weak topology. In this section we also study solutions where the initial value z 0 is different from zero.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cartan H.: Elementare Theorie der Analytischen Funktionen einer oder mehrerer Komplexen Veränderlichen. B. I.-Hochschultaschenbücher, Mannheim (1966)
Dhombres J.: Some Aspects of Functional Equations. Chulalongkorn University Press, Bangkok (1979)
Reich L., Smítal J., Štefánková M.: Local analytic solutions of the generalized Dhombres functional equations I. Sitzungsber. österr. Akad. Wiss. Wien, Math.-nat Kl. Abt. II 214, 3–25 (2005)
Reich L., Smítal J., Štefánková M.: Local analytic solutions of the generalized Dhombres functional equations II. J. Math. Anal. Appl. 355, 821–829 (2009)
Siegel C.L.: Vorlesungen über Himmelsmechanik. Springer, Berlin (1956)
Treves F.: Topological Vector Spaces, Distributions and Kernels. Academic Press, New York-London (1967)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Reich, L., Tomaschek, J. Some Remarks to the Formal and Local Theory of the Generalized Dhombres Functional Equation. Results. Math. 63, 377–395 (2013). https://doi.org/10.1007/s00025-011-0203-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-011-0203-0