A generalization of even and odd functions (Q656021)

Consider the \(\mathbb{C}\)-space \(F\) of all complex functions of a complex variable and fix an integer \(n>1\). For each integer \(r\) with \(0\leq r<n\) the authors define a function \(f\) \(\in F\) to be of type \(r\) mod \(n\) if \(f(\varepsilon z)=\varepsilon^{r}f(z)\) for all \(z\in\mathbb{C}\) and each \(n\)th root \(\varepsilon\) of \(1\). (Clearly it is sufficient to check the condition for one primitive \(n\)th root \(\varepsilon\) of \(1\).) Even and odd functions are of type \(0\) mod \(2\) and \(1\) mod \(2\), respectively, and the object of the paper is to show how known properties of even and odd functions generalize. If \(\varepsilon\) is a primitive \(n\)th root of \(1\), then for any function \(f\) the functions \[ f_{r}(z):=\frac{1}{n}\sum_{k=0}^{n-1}\varepsilon^{-kr}f(\varepsilon ^{k}z)\text{ for }r=0,1,\dots ,n-1 \] are of type \(r\) mod \(n\) and their values are independent of the choice of \(\varepsilon\). If \(F_{r}\) denotes the subspace of \(F\) consisting of all functions of type \(r\) mod \(n\) then \(F=F_{0}\oplus\dots\oplus F_{n-1}\) and \(f\longmapsto f_{r}\) is the projection of \(F\) onto \(F_{r}\). If we restrict to analytic functions in \(F\), then differentiation permutes these subspaces cyclically. The paper concludes by showing a link between these ideas and representations of finite groups.