# The construction of G-invariant finite tight frames

## Richard Vale and Shayne Waldron

## Abstract:

We give a complete classification of the
finite tight frames which are
$G$--invariant,
i.e., invariant under the unitary action of group $G$.
This result is constructive,
and we use it to consider a number of examples. In particular,
we determine the minimum number of generators for a tight frame
for the orthogonal polynomials on an $n$--gon or cube,
which is invariant under the symmetries of the weight.

**Keywords:**
Finite tight frame,
$G$--invariant frame,
group frame,
harmonic frame,
group representation,
linear action,
complex reflection group,
orthgonal polynomials on a regular polygon,
orthgonal polynomials on a cube,

**Math Review Classification:**
Primary 13A50, 33C52, 42C05, 42C15;
Secondary 41A10, 52B11, 52B15

**Length:** 23 pages

**Last Updated:** 20 January 2015

## Availability:

- pdf
- pdf (J. Fourier Anal. Appl.)