Tight frames and their symmetries

Richard Vale and Shayne Waldron


The aim of this paper is to investigate symmetry properties of tight frames, with a view to constructing tight frames of orthogonal polynomials in several variables which share the symmetries of the weight function, and other similar applications. This is achieved by using representation theory to give methods for constructing tight frames as orbits of groups of unitary transformations acting on a given finite-dimensional Hilbert space. Along the way, we show that a tight frame is determined by its Gram matrix and discuss how the symmetries of a tight frame are related to its Gram matrix. We also give a complete classification of those tight frames which arise as orbits of an abelian group of symmetries.

Keywords: Tight frames, isometric tight frames, Gram matrix, multivariate orthogonal polynomials, symmetry groups, harmonic frames, representation theory, wavelets

Math Review Classification: Primary 05B20, 33C50, 20C15, 42C15 ; Secondary 52B15, 42C40i

Length: 30 pages

Last Updated: 2 February 2004