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arXiv · 1303.1159

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
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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.

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Martin S. Copenhaver, Yeon Hyang Kim, Cortney Logan, Kyanne Mayfield, Sivaram K. Narayan, Matthew J. Petro, Jonathan Sheperd. 2013-03-05. Diagram vectors and Tight Frame Scaling in Finite Dimensions. https://arxiv.org/abs/1303.1159

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