Tight frames for cyclotomotic fields and other rational vector spaces

Tuan-Yow Chien, Victor Flynn and Shayne Waldron


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