On functions with zero mean over a finite group
Let \(M\) be a finite measure space and let \(f:M \to {\mathbb R}\) be a measurable function. Denote by \(m(f)\) the relative measure of the set \(\{x\in M: f(x)\leq 0\}\) and by \(\overline f_{G}\) the average of \(f\) over the action of a finite group \(G\) of measure-preserving transformations of \(M\), i.e., \(\overline f_{G}:=| G| ^{-1}\sum_{g\in G}f(gx)\). The main results of the paper describe lower bounds for \(m(f)\) in terms of \(G\), for example, it is shown that \(m(f)\geq| G| ^{-1}\) if \(\overline f_{G}\equiv 0\). For a particular case in which \(M=S^{1}\) with Lebesgue measure and the cyclotomic group \(G\) of rotations of order \(n\) it is shown that \(m(f)\geq 1/n\) for any trigonometric polynomial of degree less than \(n\) with zero mean. This result is generalized also for the case of functions of several variables.
scientific article; zbMATH DE number 1179573