Fourier transforms and bent functions on finite groups

Let G be a finite nonabelian group. Bent functions on G are defined by the Fourier transforms at irreducible representations of G. We introduce a dual basis , consisting of functions on G determined by its unitary irreducible representations, that will play a role similar to the dual group of a finite abelian group. Then we define the Fourier transforms as functions on , and obtain characterizations of a bent function by its Fourier transforms (as functions on ). For a function f from G to another finite group, we define a dual function on , and characterize the nonlinearity of f by its dual function . Some known results are direct consequences. Constructions of bent functions and perfect nonlinear functions are also presented.