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Decompositions of frames and a new frame identity

Decompositions of frames and a new frame identity -...

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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 f 2 i I | f, f i | 2 , respectively, i I | f, f i | 2 B f 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 = 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 equal-norm frame and if the frame elements have norm 1 it is called a unit-norm 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 ) = i I f, f i e i . Hence, T ( f ) 2 = i I | f, f i | 2 . The frame operator for the frame is the positive, self-adjoint invertible operator S = T T : H H satisfying Sf = i I f, f i f i , for all f H . Further author information: (Send correspondence to P.G. Casazza) R.B.: E-mail: [email protected] P.G.C.: E-mail: [email protected] D.E.: E-mail: [email protected] G.K.: E-mail: [email protected]
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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 i I | a i | 2 1 / 2 i I a i f i B i I | a i | 2 1 / 2 . If { f i } i I also spans H it is called a Riesz basis for H . For the fundamentals of frame theory we refer the reader to Christensen. 14 A fundamental question in frame theory involves understanding the behavior of subsets of a Parseval frame.
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