Diagram vectors and tight frame scaling in finite dimensions

Martin S. Copenhaver, Yeon Hyang Kim, Cortney Logan, Kyanne Mayfield, Sivaram K. Narayan, Matthew J. Petro, Jonathan Sheperd

Research output: Contribution to journalArticlepeer-review

15 Scopus citations

Abstract

We consider frames in a finite-dimensional Hilbert space Hn where frames are exactly the spanning sets of the vector space. The diagram vector of a vector in ℝ2 was previously defined using polar coordinates and was used to characterize tight frames in ℝ2 in a geometric fashion. Reformulating the definition of a diagram vector in ℝ2 we provide a natural extension of this notion to ℝn and ℂn. Using the diagram vectors we give a characterization of tight frames in ℝn or ℂn. Further we provide a characterization of when a unit-norm frame in ℝn or ℂn can be scaled to a tight frame. This classification allows us to determine all scaling coefficients that make a unit-norm frame into a tight frame.

Original languageEnglish
Pages (from-to)73-88
Number of pages16
JournalOperators and Matrices
Volume8
Issue number1
DOIs
StatePublished - Mar 2014

Keywords

  • Diagram vectors
  • Frames
  • Gramian operator
  • Tight frame scaling
  • Tight frames

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