Title:
Approximation by Group Invariant Subspaces
Abstract:
In this talk we will look at the structure and approximation properties of $\Gamma$-invariant
spaces of $L^2(R)$, where $R$ is a second countable LCA group. The invariance is with
respect to the action of $\Gamma$, a non commutative group in the form of a semidirect
product of a discrete cocompact subgroup of $R$ and a group of automorphisms. This
class includes in particular most of the crystallographic groups.
By defining a range function we are able to characterize the property of being invariant under the action of the crystal-group with a property of the range function. We then show how these results can be applied to prove the existence and construction of a $\Gamma$-invariant subspace that best approximates a set of functional data in $L^2(R)$. This is very relevant in applications since in the euclidean case, $\Gamma$-invariant subspaces are invariant under rigid movements, a very sought feature in models for signal processing. |