Constructions of biangular tight frames and their relationships with equiangular tight frames. (English) Zbl 1398.42021
Kim, Yeonhyang (ed.) et al., Frames and harmonic analysis. AMS special session on frames, wavelets and Gabor systems and special session on frames, harmonic analysis, and operator theory, North Dakota State University, Fargo, ND, USA, April 16–17, 2016. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-3619-3/pbk; 978-1-4704-4723-6/ebook). Contemporary Mathematics 706, 1-19 (2018).
Summary: 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 (of minimal coherence).
We develop a general framework of so-called Steiner BTFs – which includes the well-known Steiner ETFs as special cases; surprisingly, the development of this framework leads to a connection with famously open problems regarding the existence of Mersenne and Fermat primes.
In addition, we demonstrate an example of a smooth parametrization of \(6\)-vector BTFs in \(\mathbb R^3\), where the curve “passes through” an ETF; moreover, the corresponding frame angles “deform” smoothly with the parametrization, thereby answering two questions about the rigidity of BTFs.
Finally, we generalize from BTFs to (chordally) biangular tight fusion frames (BTFFs) – basis-like sets of orthogonal projections admitting exactly two distinct trace inner products – and we explain how one may think of them as generalizations of BTFs. In particular, we construct an interesting example of a BTFF corresponding to \(16\) \(2\)-dimensional subspaces of \(\mathbb R^4\) that “Plücker embeds” into a Steiner ETF consisting of \(16\) vectors in \(\mathbb R^6\), which we refer to as a Plücker ETF.
For the entire collection see [Zbl 1390.42001].
We develop a general framework of so-called Steiner BTFs – which includes the well-known Steiner ETFs as special cases; surprisingly, the development of this framework leads to a connection with famously open problems regarding the existence of Mersenne and Fermat primes.
In addition, we demonstrate an example of a smooth parametrization of \(6\)-vector BTFs in \(\mathbb R^3\), where the curve “passes through” an ETF; moreover, the corresponding frame angles “deform” smoothly with the parametrization, thereby answering two questions about the rigidity of BTFs.
Finally, we generalize from BTFs to (chordally) biangular tight fusion frames (BTFFs) – basis-like sets of orthogonal projections admitting exactly two distinct trace inner products – and we explain how one may think of them as generalizations of BTFs. In particular, we construct an interesting example of a BTFF corresponding to \(16\) \(2\)-dimensional subspaces of \(\mathbb R^4\) that “Plücker embeds” into a Steiner ETF consisting of \(16\) vectors in \(\mathbb R^6\), which we refer to as a Plücker ETF.
For the entire collection see [Zbl 1390.42001].
MSC:
| 42C15 | General harmonic expansions, frames |
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