Optimal Parseval frames: total coherence and total volume

We introduce three quantities called total coherence, total volume, and nuclear energy, and we show that equiangular Parseval frames maximize all three of these quantities over the set of all Parseval frames. We then show that equiangular Parseval frames also maximize the total volume and nuclear en...

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Bibliographic Details
Published in:Linear & multilinear algebra 2023-09, Vol.71 (13), p.2067-2092
Main Authors: Cahill, Jameson, Casazza, Peter G.
Format: Article
Language:English
Subjects:
Coherence
equiangular
Frame theory
Frames
Nuclear energy
Nuclear reactor components
Nuclear reactors
unit norm tight frames
Welch bound
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Summary:We introduce three quantities called total coherence, total volume, and nuclear energy, and we show that equiangular Parseval frames maximize all three of these quantities over the set of all Parseval frames. We then show that equiangular Parseval frames also maximize the total volume and nuclear energy over the set of equal norm frames. Along the way, we derive a bound on the smallest k-dimensional volume in an equal norm frame which is a generalization of the Welch bound. We conclude with an extensive list of open questions.
ISSN:0308-1087
1563-5139
DOI:10.1080/03081087.2022.2093320