On the number of harmonic frames
Simon Marshall and Shayne Waldron
Abstract:
There is a finite number $h_{n,d}$
of tight frames of $n$ distinct vectors
for $\C^d$ which are the orbit of a
vector under a unitary action of
the cyclic group $\Z_n$.
These cyclic harmonic frames
(or geometrically uniform tight frames
are used in signal analysis
and quantum information theory,
and provide many tight frames of particular interest.
Here we investigate the
conjecture that
$h_{n,d}$
grows like $n^{d-1}$.
By using a result of Laurent which describes
the set of solutions of algebraic equations in roots of unity,
we prove the asymptotic estimate
$$ h_{n,d} \approx {n^d \over \varphi(n)}\ge n^{d-1}, \qquad n\to\infty. $$
By using a group theoretic approach,
we also give some exact formulas for $h_{n,d}$, and estimate
the number of cyclic harmonic frames up to projective unitary equivalence.
Keywords:
Finite tight frames,
harmonic frames,
finite abelian groups,
character theory,
Pontryagin duality,
Mordell-Lang conjecture,
projective unitary equivalence
Math Review Classification:
Primary 42C15, 94A12;
Secondary 11Z05, 20C15, 94A15,
20F28
Length: 22 pages
Last Updated: 22 November 2016
Availability: