caa a22 4500 463247027 CHVBK 20180405153328.0 cr unu---uuuuu 170326e20071001xx s 000 0 eng 10.1007/s00041-006-6043-8 doi (NATIONALLICENCE)springer-10.1007/s00041-006-6043-8 The Uniqueness of Shift-Generated Duals for Frames in Shift-Invariant Subspaces [Elektronische Daten] [A. Askari Hemmat, Jean-Pierre Gabardo] 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. Birkhauser Boston, 2007 Hemmat A. Askari Department of Mathematics, Vali-Asr University of Rafsanjan, Iran aut Gabardo Jean-Pierre Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada aut Journal of Fourier Analysis and Applications Birkhäuser-Verlag; www.birkhauser.com 13/5(2007-10-01), 589-606 1069-5869 13:5<589 2007 13 41 https://doi.org/10.1007/s00041-006-6043-8 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00041-006-6043-8 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Hemmat A. Askari Department of Mathematics, Vali-Asr University of Rafsanjan, Iran aut NATIONALLICENCE 700 1- Gabardo Jean-Pierre Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada aut NATIONALLICENCE 773 0- Journal of Fourier Analysis and Applications Birkhäuser-Verlag; www.birkhauser.com 13/5(2007-10-01), 589-606 1069-5869 13:5<589 2007 13 41 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer