Triple Correlation - Extension To Groups

Extension To Groups

The triple correlation may be defined for any locally compact group by using the group's left-invariant Haar measure. It is easily shown that the resulting object is invariant under left translation of the underlying function and unbiased in additive Gaussian noise. What is more interesting is the question of uniqueness : when two functions have the same triple correlation, how are the functions related? For many cases of practical interest, the triple correlation of a function on an abstract group uniquely identifies that function up to a single unknown group action. This uniqueness is a mathematical result that relies on the Pontryagin duality theorem, the Tannaka-Krein duality theorem, and related results of Iwahori-Sugiura, and Tatsuuma. Algorithms exist for recovering bandlimited functions from their triple correlation on euclidean space, as well as rotation groups in two and three dimensions. There is also an interesting link with Wiener's tauberian theorem: any function whose translates are dense in, where G is a locally compact abelian group, is also uniquely identified by its triple correlation.