  For every natural number k we introduce the notion of k-th order convolution
of functions on abelian groups. We study the group of convolution preserving
automorphisms of function algebras in the limit. It turns out that such groups
have k-nilpotent factor groups explaining why k-th order Fourier analysis has
non-commutative features. To demonstrate our method in the quadratic case we
develop a new quadratic representation theory on finite abelian groups. We
introduce the notion of a quadrtic nil-morphism of an abelian group into a two
step nil-manifold. We prove a structure theorem saying that any bounded
function on a finite abelian group is decomposable into a structured part
(which is the composition of a nil-morphism with a bounded complexity
continuous function) and a random looking part with small U3 norm. It implies a
new inverse theorem for the U3 norm. (The general case for Un, n>3 will be
discussed in the next part of this sequence.) We point out that our framework
creates interesting limit objects for functions on finite (or compact) abelian
groups that are measurable functions on nil-manifolds.
