Minimal scalings and structural properties of scalable frames

Alice Chan, Rachel Domagalski, Yeon Hyang Kim, Sivaram K. Narayan, Hong Suh, Xingyu Zhang

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish
Pages (from-to)1057-1073
Number of pages17
JournalOperators and Matrices
Volume11
Issue number4
DOIs
StatePublished - Dec 2017

Keywords

  • Diagram vectors
  • Frames
  • Minimal scalings
  • Scalable frames
  • Tight frames

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