TY - JOUR

T1 - Minimal scalings and structural properties of scalable frames

AU - Chan, A

AU - Narayan, Sivaram Krishnan

N1 - Funding Information:
Acknowledgement. Much of this work was done when R. Domagalski, H. Suh, and X. Zhang participated in the Central Michigan University NSF-REU program in the summer of 2014. A. Chan was the graduate student mentor. Kim was supported by the Central Michigan University FRCE Research Type A Grant #C48143.
Funding Information:
Research supported by NSF-REU grant DMS 11-56890.
Funding Information:
Research supported by NSFNational Science Foundation -REU grant DMS 11-56890. Kim was supported by the Central Michigan UniversityCMU FRCE Research Type A Grant #C48143.
Publisher Copyright:
© 2017, Element D.O.O. All rights reserved.

PY - 2017

Y1 - 2017

N2 - For a unit-norm frame F = (fi)ki=1 in ℝn, a scaling is a vector c = (c(1),…,c(k)) ∈ ℝk≥0 such that (√c(i) fi)ki=1 is a Parseval frame in ℝn. If such a scaling exists, F is said to be scalable. A scaling c is a minimal scaling if (fi: 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)) ∈ ℝk>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.

AB - For a unit-norm frame F = (fi)ki=1 in ℝn, a scaling is a vector c = (c(1),…,c(k)) ∈ ℝk≥0 such that (√c(i) fi)ki=1 is a Parseval frame in ℝn. If such a scaling exists, F is said to be scalable. A scaling c is a minimal scaling if (fi: 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)) ∈ ℝk>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.

M3 - Article

SN - 1846-3886

VL - 11

SP - 1057

EP - 1073

JO - Operators and Matrices

JF - Operators and Matrices

ER -