Fourier transforms and bent functions on faithful actions of finite abelian groups

Let G be a finite abelian group acting faithfully on a finite set X. The G-bentness and G-perfect nonlinearity of functions on X are studied by Poinsot and co-authors (Discret Appl Math 157:1848–1857, 2009; GESTS Int Trans Comput Sci Eng 12:1–14, 2005) via Fourier transforms of functions on G. In this paper we introduce the so-called \(G\)-dual set \(\widehat{X}\) of X, which plays the role similar to the dual group \(\widehat{G}\) of G, and develop a Fourier analysis on X, a generalization of the Fourier analysis on the group G. Then we characterize the bentness and perfect nonlinearity of functions on X by their own Fourier transforms on \(\widehat{X}\). Furthermore, we prove that the bentness of a function on X can be determined by its distance from the set of G-linear functions. As direct consequences, many known results in Logachev et al. (Discret Math Appl 7:547–564, 1997), Carlet and Ding (J Complex 20:205–244, 2004), Poinsot (2009), Poinsot et al. (2005) and some new results about bent functions on G are obtained. In order to explain the theory developed in this paper clearly, examples are also presented.