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The Uniqueness of Shift-Generated Duals for Frames in Shift-Invariant Subspaces

Authors:	A Askari Hemmat Jean-Pierre Gabardo

Institution:	(1) Department of Mathematics, Vali-Asr University of Rafsanjan, Iran;(2) Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada

Abstract:	Given an invertible $n\times n$ matrix B and $\Phi$ a finite or countable subset of $L^2(\mathbb{R}^n)$ , we consider the collection $X=\{\phi(\cdot -Bk): \,\phi\in\Phi,\,k\in \mathbb{Z}^n\}$ generating the closed subspace $\mathcal{M}$ of $L^2(\mathbb{R}^n)$ . If that collection forms a frame for $\mathcal{M}$ , one can introduce two different types of shift-generated (SG) dual frames for X, called type I and type II SG-duals, respectively. The main distinction between them is that a SG-dual of type I is required to be contained in the space $\mathcal{M}$ generated by the original frame while, for a type II SG-dual, one imposes that the range of the frame transform associated with the dual be contained in the range of the frame transform associated with the original frame. We characterize the uniqueness of both types of duals using the Gramian and dual Gramian operators which were introduced in an article by Ron and Shen and are known to play an important role in the theory of shift-invariant spaces.

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