Maximum robustness and surgery of frames in finite dimensions

Copenhaver, Martin S.; Kim, Yeon Hyang; Logan, Cortney; Mayfield, Kyanne; Narayan, Seema; Sheperd, Jonathan

doi:10.1016/j.laa.2013.04.016

Cited by 7 publications

(5 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The diagram vectors give us the following characterizations of tight frames and scalable frames: Theorem 7. [17,16] Let {f i } k i=1 be a sequence of vectors in H, not all of which are zero. Then {f i } k i=1 is a tight frame if and only if k i=1 fi = 0.…”

Section: 1mentioning

confidence: 99%

See 1 more Smart Citation

Scalability of Frames Generated by Dynamical Operators

Aceska¹,

Kim²

2017

Front. Appl. Math. Stat.

View full text Add to dashboard Cite

Let H be a separable Hilbert space, let G ⊂ H, and let A be an operator on H. Under appropriate conditions on A and G, it is known that the set of iterationsis a frame for H. We call F G (A) a dynamical frame for H, and explore further its properties; in particular, we show that the canonical dual frame of F G (A) also has an iterative set structure. We explore the relations between the operator A, the set G and the number of iterations L which ensure that the system F G (A) is a scalable frame. We give a general statement on frame scalability, and study in detail the case when A is a normal operator, utilizing the unitary diagonalization. In addition, we answer the question of when F G (A) is a scalable frame in several special cases involving block-diagonal and companion operators.

show abstract

Section: 1mentioning

confidence: 99%

“…Theorem 8. [17,16] Let {f i } k i=1 be a unit-norm frame for H and c 1 , • • • , c k be nonnegative numbers, which are not all zero. Let G be the Gramian associated to the diagram vectors…”

Section: 1mentioning

confidence: 99%

Scalability of Frames Generated by Dynamical Operators

Aceska¹,

Kim²

2017

Front. Appl. Math. Stat.

View full text Add to dashboard Cite

show abstract

“…Theorem 2.1 ( [10,9]). Let {f i } k i=1 be a sequence of vectors in R n , not all of which are zero.…”

Section: Preliminariesmentioning

confidence: 99%

“…Moreover, Theorem 4.10 gives conditions for c under which the empty cover of cF is pairwise disjoint. That is, if we have two different collection of subsets of minimal scalings for the orthogonal decomposition (10), then EC(cF ) is not pairwise disjoint. We note the orthogonal decomposition (10) is not unique in general.…”

Section: Theorem 48 ([7]mentioning

confidence: 99%

See 1 more Smart Citation

Minimal scalings and structural properties of scalable frames

Chan

Domagalski

Kim³

et al. 2017

Operators and Matrices

Self Cite

View full text Add to dashboard Cite

Abstract. For a unit-norm frame F = { f i } k i=1 in R n , a scaling is a vector c = (c(1),... ,c(kis a Parseval frame in R n . If such a scaling exists, F is said to be scalable. A scaling c is a minimal scaling if { f i : c(i) > 0} has no proper scalable subframe. In this paper, we provide an algorithm to find all possible contact points for the John's decomposition of the identity by applying the b-rule algorithm to a linear system which is associated with a scalable frame. We also give an estimate of the number of minimal scalings of a scalable frame. We provide a characterization of when minimal scalings are affinely dependent. Using this characterization, we can conclude that all strict scalings c = (c(1),... ,c(k)) ∈ R k >0 of F have the same structural property. That is, the collections of all tight subframes of strictly scaled frames are the same up to a permutation of the frame elements. We also present the uniqueness of orthogonal partitioning property of any set of minimal scalings, which provides all possible tight subframes of a given scaled frame. Mathematics subject classification (2010): Primary 42C15, 05B20, 15A03.

show abstract