Introduction to Finite Frame Theory - Frame Research Center

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28 Peter G. Casazza, Gitta Kutyniok, and Friedrich Philipp 5.2 Properties of Dual Frames While focussing on properties of the canonical dual frame in the last subsection, we next discuss properties shared by all dual frames. The first question arising is: How do you characterize all dual frames? A comprehensive answer is provided by the following result. Proposition 17. Let (ϕ i ) M i=1 be a frame for H N with analysis operator T and frame operator S. Then the following conditions are equivalent. (i) (ψ i ) M i=1 is a dual frame for (ϕ i) M i=1 . (ii) The analysis operator T 1 of the sequence (ψ i − S −1 ϕ i ) M i=1 satisfies ranT ⊥ ran T 1 . Proof. We set ˜ϕ i := ψ i − S −1 ϕ i for i = 1,...,M and note that M ∑ i=1 〈x,ψ i 〉ϕ i = M ∑ i=1 〈x, ˜ϕ i + S −1 ϕ i 〉ϕ i = x + M ∑ i=1 〈x, ˜ϕ i 〉ϕ i = x + T ∗ T 1 x holds for all x ∈ H N . Hence, (ψ i ) M i=1 is a dual frame for (ϕ i) M i=1 T ∗ T 1 = 0. But this is equivalent to (ii). ⊓⊔ if and only if From this result, we have the following corollary which provides a general formula for all dual frames. Corollary 9. Let (ϕ i ) M i=1 be a frame for H N with analysis operator T and frame operator S with associated normalized eigenvectors (e j ) N j=1 and respective eigenvalues (λ j ) N j=1 . Then every dual frame {ψ i} M i=1 for (ϕ i) M i=1 is of the form ψ i = N ∑ j=1 ( 1 λ j 〈ϕ i ,e j 〉 + h i j ) e j , i = 1,...,M, where each (h i j ) M i=1 , j = 1,...,N, is an element of (ranT )⊥ . Proof. If ψ i , i = 1,...,M, is of the given form with sequences (h i j ) M i=1 ∈ lM 2 , j = 1,...,N, then ψ i = S −1 ϕ i + ˜ϕ i , where ˜ϕ i := ∑ N j=1 h i je j , i = 1,...,M. The analysis operator ˜T of ( ˜ϕ i ) M i=1 satisfies ˜T e j = (h i j ) M i=1 . The claim follows from this observation. ⊓⊔ As a second corollary, we derive a characterization of all frames which have a uniquely determined dual frame. It is evident, that this unique dual frame coincides with the canonical dual frame. Corollary 10. A frame (ϕ i ) M i=1 for H N has a unique dual frame if and only if M = N.
Introduction to Finite Frame Theory 29 5.3 Frame Algorithms Let (ϕ i ) M i=1 be a frame for H N with frame operator S, and assume we are given the image of a signal x ∈ H N under the analysis operator, i.e., the sequence (〈x,ϕ i 〉) M i=1 in l M 2 . Theorem 8 already provided us with the reconstruction formula x = M ∑ i=1 〈x,ϕ i 〉S −1 ϕ i by using the canonical dual frame. Since inversion is typically not only computationally expensive, but also numerically instable, this formula might not be utilizable in practice. To resolve this problem, we will next discuss three iterative methods to derive a converging sequence of approximations of x from knowledge of (〈x,ϕ i 〉) M i=1 . The first on our list is the so-called frame algorithm. Proposition 18 (Frame Algorithm). Let (ϕ i ) M i=1 be a frame for H N with frame bounds A,B and frame operator S. Given a signal x ∈ H N , define a sequence (y j ) ∞ j=0 in H N by y 0 = 0, y j = y j−1 + 2 A + B S(x − y j−1) for all j ≥ 1. Then (y j ) ∞ j=0 converges to x in H N and the rate of convergence is ‖x − y j ‖ ≤ Proof. First, for all x ∈ H N , we have 〈( Id − 2 ) 〉 A + B S x,x = ‖x‖ 2 − 2 A + B Similarly, we obtain which yields By the definition of y j , for any j ≥ 0, ( ) B − A j ‖x‖, j ≥ 0. B + A M ∑ i=1 |〈x,ϕ i 〉| 2 ≤ ‖x‖ 2 − 2A A + B ‖x‖2 = B − A A + B ‖x‖2 . − B − A 〈( B + A ‖x‖2 ≤ Id − 2 ) 〉 A + B S x,x , ∥ ∥Id − 2 A + B S ∥ ∥∥ ≤ B − A A + B . (3) x − y j = x − y j−1 − 2 ( A + B S(x − y j−1) = Id − 2 ) A + B S (x − y j−1 ). Iterating this calculation, we derive
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