Decompositions of frames and a new frame identity

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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 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 ) = 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, self-adjoint 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.: E-mail: radu.balan@siemens.com P.G.C.: E-mail: pete@math.missouri.edu D.E.: E-mail: edidin@math.missouri.edu G.K.: E-mail: gitta.kutyniok@math.uni-giessen.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 !...
View Full Document

Page1 / 10

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online