Minimal scalings and structural properties of scalable frames

For a unit-norm frame F = (f_i)^k_i=1 in ℝⁿ, a scaling is a vector c = (c(1),…,c(k)) ∈ ℝ^k_≥0 such that (√c(i) f_i)^k_i=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 (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.