The Structure of Finitely Generated Shift-Invariant Subspaces on Locally Compact Abelian Groups

Abstract

In this paper, we characterize finitely generated shift-invariant subspaces of $$L^2(G)$$ , where G is a locally compact abelian group. In particular, we give a formula for the coefficients in the known representation of the Fourier transform of the elements of finitely generated shift-invariant subspaces. Also, certain orthogonalization procedure for generators which is reminiscent of the Gram–Schmidt orthogonalization process is given.