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Diagram vectors and tight frame scaling in finite dimensions. (English) Zbl 1435.42026

Summary: We consider frames in a finite-dimensional Hilbert space \(\mathcal H_n\) where frames are exactly the spanning sets of the vector space. The diagram vector of a vector in \(\mathbb R^2\) was previously defined using polar coordinates and was used to characterize tight frames in \(\mathbb R^2\) in a geometric fashion. Reformulating the definition of a diagram vector in \(\mathbb R^2\) we provide a natural extension of this notion to \(\mathbb R^n\) and \( \mathbb C^n\). Using the diagram vectors we give a characterization of tight frames in \(\mathbb R^n\) or \(\mathbb C^n\). Further we provide a characterization of when a unit-norm frame in \(\mathbb R^n\) or \(\mathbb C^n\) 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.

MSC:

42C15 General harmonic expansions, frames
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
15A03 Vector spaces, linear dependence, rank, lineability
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