Minimal scalings and structural properties of scalable frames. (English) Zbl 1378.42017

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

Oper. Matrices 11, No. 4, 1057-1073 (2017).

Summary: For a unit-norm frame \(F=\{f_i\}^k_{i=1} \mathrm{in} \mathbb{R}^{n}\), a scaling is a vector \(c= (c(1),\ldots,c(k))\in \mathbb{R}^{k}_{\geqslant 0}\) such that \(\{\sqrt{c(i)f_i}\}^k_{i=1}\) is a Parseval frame in \(\mathbb{R}^{n}\). 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 \(c= (c(1),\ldots,c(k)) \in \mathbb{R}^{k}_{>0}\) 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 the orthogonal partitioning property of any set of minimal scalings, which provides all possible tight subframes of a given scaled frame.

MSC:

42C15	General harmonic expansions, frames
05B20	Combinatorial aspects of matrices (incidence, Hadamard, etc.)
15A03	Vector spaces, linear dependence, rank, lineability

Keywords:

frames; tight frames; diagram vectors; scalable frames; minimal scalings