Tight frames for cyclotomic fields and other rational vector spaces

Here we consider the construction of tight frames for rational vector spaces. This is a subtle question, because the inner products on Qd are not all isomorphic. We show that a tight frame for Cd 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 a tight frame with a given Gramian. We then consider the rational vector space given the cyclotomic field Q(ω), with ω a primitive n-th root of unity. We give a simple formula for the unique inner product which makes the n-th roots 1,ω,ω2,…,ωn−1 into a tight frame for Q(ω). From this, we conclude that the associated “canonical coordinates” have many nice properties, e.g., multiplication in Q(ω) 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.