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Unformatted text preview: Decompositions of frames and a new frame identity Radu Balan a , Peter G. Casazza b , Dan Edidin c and Gitta Kutyniok d a Siemens Corporate Research, 755 College Road East, Princeton, NJ 08540, USA; b Department of Mathematics, University of Missouri, Columbia, MO 65211, USA; c Department of Mathematics, University of Missouri, Columbia, MO 65211, USA; d Mathematical Institute, Justus–Liebig–University Giessen, 35392 Giessen, Germany ABSTRACT We analyze a fundamental question in Hilbert space frame theory: What is the optimal decomposition of a Parseval frame? We will see that this question impacts several famous unsolved problems in different areas of mathematics. As a step towards the solution of this question, we give a new identity which holds for all Parseval frames. Keywords: frame, Riesz basis, Bessel sequence 1. INTRODUCTION Let H be a Hilbert space and f i ∈ H for i ∈ I . Let K = span { f i } i ∈ I . A number A > 0 (respectively, B > 0) is called a lower (respectively, upper ) frame bound for F = { f i } i ∈ I if for all f ∈ K A k f k 2 ≤ X i ∈ I h f, f i i 2 , respectively, X i ∈ I h f, f i i 2 ≤ B k f k 2 . If B < ∞ we call F = { f i } i ∈ I a Bessel sequence with Bessel bound B . If 0 < A ≤ B < ∞ , then { f i } i ∈ I is a frame for K . If K 6 = H we call { f i } i ∈ I a frame sequence in H . The largest A and the smallest B satisfying the above inequalities are called the optimal lower and upper frame bound and will be denoted A ( F ) and B ( F ) respectively. If A = B = λ we call this a λ tight frame and if λ = 1 it is called a Parseval frame . If all the frame elements have the same norm we call this an equalnorm frame and if the frame elements have norm 1 it is called a unitnorm frame . If { f i } i ∈ I is a Bessel sequence, the synthesis operator for { f i } i ∈ I is the bounded linear operator T : ` 2 ( I ) → H given by T ( e i ) = f i for all i ∈ I where { e i } i ∈ I is the unit vector basis of ` 2 ( I ). The analysis operator for { f i } i ∈ I is T ∗ and satisfies: T ∗ ( f ) = X i ∈ I h f, f i i e i . Hence, k T ∗ ( f ) k 2 = X i ∈ I h f, f i i 2 . The frame operator for the frame is the positive, selfadjoint invertible operator S = T T ∗ : H → H satisfying Sf = X i ∈ I h f, f i i f i , for all f ∈ H . Further author information: (Send correspondence to P.G. Casazza) R.B.: Email: radu.balan@siemens.com P.G.C.: Email: pete@math.missouri.edu D.E.: Email: edidin@math.missouri.edu G.K.: Email: gitta.kutyniok@math.unigiessen.de Finally, the family { f i } i ∈ I is a Riesz basic sequence in H with Riesz basis bounds A, B if for all sequences of scalars { a i } i ∈ I we have A X i ∈ I  a i  2 !...
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 Spring '10
 RunyiYu

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