Functional equations on finite groups of substitutions
Motivated by Babbage, the authors study the functional equation \[ F(f\circ g_1(t), \dots, f\circ g_r(t), t) =0, \tag{1} \] where the functions \(g_1, \dots, g_r\) and \(F\) are given and \(f\) is to be determined. However, under such general circumstances, there is no hope to give the set of solutions. As Babbage illustrates it via several examples, it may occur that, having a particular solution, infinitely many functions \(f\) can be created satisfying the functional equation (1), although the family of such functions may not be the complete collection of all solutions, see [\textit{C. G. Small}, Functional equations and how to solve them. Problem Books in Mathematics. New York, NY: Springer (2007; Zbl 1109.39022)]. The problem becomes more tractable if the functions \(g_1, \dots, g_r\) form a group under composition on their domain. It turns out that the algebraic structure of a stabilizer determines the number of initial value conditions for the functional equation. The first systematic investigations in this direction are due to \textit{L. Presić} [Publ. Fac. Electrotech. Univ. Belgrade, Ser. Math. Phys. 115--121, 21--28 (1963; Zbl 0166.41202)], who characterized all solutions of (1) when \(F\) is linear and does not depend on its last variable. The authors give examples and the method given in the examples is applied in more general settings. Composing both sides of (1) by the elements of the group, a system of nonlinear equations is obtained where \(\Phi_i = f\circ g_i\) play the role of unknowns. Then, under additional regularity assumptions, the implicit function theorem offers a possible approach to provide a solution. Such an approach was presented by \textit{M. Bessenyei} [Am. Math. Mon. 117, No. 10, 921--927 (2010; Zbl 1208.39032)] under the assumptions that the group is an abelian one and its elements have a common fixed point in the interior of the domain. But the paper concerns only a special case of (1). The general form of (1) was investigated by \textit{M. Bessenyei} and \textit{C. G. Kézi} [Linear Algebra Appl. 434, No. 6, 1525--1531 (2011; Zbl 1222.39018)], where the abelian structure is omitted but stronger regularity is required. The assumption on the existence of a common fixed point is kept. The required regularity properties, given by [Bessenyei, 2010, loc. cit.] and by [Bessenyei, Kézi, 2011, loc. cit.], turn out to be quite restrictive in some sense. The aim of the paper is to handle these problems in a unified manner that generalizes the previous results. The main theorem clarifies the connection of the algebraic and analytic feature of the problem, contains the previous results as special cases and offers further applications.
scientific article; zbMATH DE number 6110674
scientific article