#
Tight frames for cyclotomotic fields and other rational vector spaces

## Tuan-Yow Chien, Victor Flynn and Shayne Waldron

## Abstract:

Here we consider the construction of tight frames for rational vector spaces.
This is a subtle question, because the inner products on Q^d are
not all isomorphic.
We show that a tight frame for C^d can be arbitrarily approximated
by a *tight* frame with vectors in (Q+iQ)^d,
and hence there are *many* tight frames for
rational inner product spaces.
We investigate the ``minimal field'' for which there is
tight frame with a given Gramian.
We then consider the rational vector space given the cyclotomic
field Q(w), with w a primitive n-th root of unity.
We give a simple formula for the unique inner product which makes
the n-th roots 1,w,w^2,...,w^{n-1}
into a tight frame for Q(w).
From this, we conclude that the associated ``canonical coordinates''
have many nice properties, e.g.,
multiplication in Q(w) corresponds to convolution,
which makes them well suited to computation.
Along the way, we give a detailed description of the space of
Q-linear dependencies between the n-th roots,
which includes a cyclically invariant tight frame.

**Keywords:**
finite tight frames,
vector spaces over the rationals,
Gramian,
canonical coordinates,
Cayley transform,
least squares (minimum norm) solutions,
cyclotomic fields

**Math Review Classification:**
Primary
11R18,
15A03,
41A65,
42C15

**Length:** 24 pages

**Last Updated:** 29 October 2014

## Availability: