Minimal scalings and structural properties of scalable frames

Abstract

For a unit-norm frame F = {fi}i=1 in Rn, a scaling is a vector c = (c(1), . . . , c(k)) ∈ R≥0 such that { √ c(i)fi}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 {fi : c(i) > 0} has no proper scalable subframe. It is known that the set of all scalings of F is a convex polytope with vertices corresponding to minimal scalings. In this talk, we provide a method to find a subset of contact points which provides a decomposition of the identity, and 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>0 of F have the same structural property. 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 Talk time: 2016-07-21 02:30 PM— 2016-07-21 02:50 PM Talk location: Cupples I Room 218