## Abstract

For a unit-norm frame F = (f_{i})^{k}_{i=1} in ℝ^{n}, 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 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),…,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.

Original language | English |
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Pages (from-to) | 1057-1073 |

Number of pages | 17 |

Journal | Operators and Matrices |

Volume | 11 |

Issue number | 4 |

DOIs | |

State | Published - Dec 2017 |

## Keywords

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