Constructions of biangular tight frames and their relationships with equiangular tight frames
Abstract
We study several interesting examples of Biangular Tight Frames (BTFs) - basis-like sets of unit vectors admitting exactly two distinct frame angles (ie, pairwise absolute inner products) - and examine their relationships with Equiangular Tight Frames (ETFs) - basis-like systems which admit exactly one frame angle. We demonstrate a smooth parametrization BTFs, where the corresponding frame angles transform smoothly with the parameter, which "passes through" an ETF answers two questions regarding the rigidity of BTFs. We also develop a general framework of so-called harmonic BTFs and Steiner BTFs - which includes the equiangular cases, surprisingly, the development of this framework leads to a connection with the famous open problem(s) regarding the existence of Mersenne and Fermat primes. Finally, we construct a (chordally) biangular tight set of subspaces (ie, a tight fusion frame) which "Pl\"ucker embeds" into an ETF.
Keywords
Cite
@article{arxiv.1703.01786,
title = {Constructions of biangular tight frames and their relationships with equiangular tight frames},
author = {John I. Haas and Jameson Cahill and Janet Tremain and Peter G. Casazza},
journal= {arXiv preprint arXiv:1703.01786},
year = {2017}
}
Comments
19 pages