Journal of the
Korean Mathematical Society JKMS

Let $G$ be a metrizable, $\sigma$-compact locally compact abelian group with a compact open subgroup. In this paper we define the Gramian and the dual Gramian operators for shift invariant subspaces of $L^2(G)$ and we use them to characterize shift generated dual frames for shift invariant spaces, which forms a frame for a subspace of $L^2(G)$. We present necessary and sufficient conditions for which standard dual is a unique SG-dual frame of type I and type II.

Keywords: frames, Gramian operator, locally compact abelian group, shift invariant space, SG-dual frame

MSC numbers: Primary 42C40; Secondary 11S85