Autocorrelation Functions as Translation Invariants in L¹ and L²

Abstract

The Adler-Konheim theorem [Proc. Amer. Math. Soc. 13 (1962), 425-428] states that the collection of nth-order autocorrelation functions ${\cal M} = \{M^n(\cdot): n=1,2,\dots\}$ is a complete set of translation invariants for real-valued L¹ functions on a locally compact abelian group. It is shown here that there are proper subsets of ${\cal M}$ that also form a complete set of translation invariants, and these subsets are characterized. Specifically, a subset is complete if and only if it contains infinitely many even-order autocorrelation functions. In addition, any infinite subset of $\cal M$ is complete up to a sign. While stated here for functions on $\cal R,$ the proofs presented hold for functions on any locally compact abelian group that is not compact, in particular, on ${\cal R}^n$ and the integer lattice ${\cal Z}^n.$