Characterization of Oblique Dual Frame Pairs
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Characterization of Oblique Dual Frame Pairs Yonina C. Eldar1 and Ole Christensen2 1 Department 2 Department
of Electrical Engineering, Technion – Israel Institute of Technology, Technion City, Haifa 32000, Israel of Mathematics, Technical University of Denmark, Building DK-303, 2800 Kongens Lyngby, Denmark
Received 2 September 2004; Revised 17 December 2004; Accepted 21 January 2005 Given a frame for a subspace W of a Hilbert space H , we consider all possible families of oblique dual frame vectors on an appropriately chosen subspace V. In place of the standard description, which involves computing the pseudoinverse of the frame operator, we develop an alternative characterization which in some cases can be computationally more efficient. We first treat the case of a general frame on an arbitrary Hilbert space, and then specialize the results to shift-invariant frames with multiple generators. In particular, we present explicit versions of our general conditions for the case of shift-invariant spaces with a single generator. The theory is also adapted to the standard frame setting in which the original and dual frames are defined on the same space. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1.
INTRODUCTION
Frames are generalizations of bases which lead to redundant signal expansions [1–4]. A frame for a Hilbert space is a set of not necessarily linearly independent vectors that has the property that each vector in the space can be expanded in terms of these vectors. Frames were first introduced by Duffin and Schaeffer [1] in the context of nonharmonic Fourier series, and play an important role in the theory of nonuniform sampling [1, 2, 5, 6]. Recent interest in frames has been motivated in part by their utility in analyzing wavelet expansions [7, 8], and by their robustness properties [3, 8– 13]. Frame-like expansions have been developed and used in a wide range of disciplines. Many connections between frame theory and various signal processing techniques have been recently discovered and developed. For example, the theory of frames has been used to study and design oversampled filter banks [14–17] and error correction codes [18]. Wavelet families have been used in quantum mechanics and many other areas of theoretical physics [8, 19]. One of the prime applications of frames is that they lead to expansions of vectors (or signals) in the underlying Hilbert space in terms of the frame elements. Specifically, if H is a separable Hilbert space and { fk }∞ k=1 is a frame for H , then any f in H can be expressed as
f =
∞
k=1
f , gk fk ,
(1)
for some dual frame {gk }∞ k=1 for H . In order to use this representation in practice, we need to be able to calculate the coefficients f , gk . A popular choice of {gk }∞ k=1 is the minimal-norm dual frame, that is, the canonical dual frame. However, computing the minimal-norm dual is highly nontrivial in general. Another issue is that the frame { fk }∞ k=1 might have a certain structure which is not shared by the minimal-norm dual. This com
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