Full Spark Frames in the Orbit of a Representation

TL;DR

This paper introduces a new infinite family of full spark frames derived from a unitary group representation involving a semi-direct product, expanding beyond previously known harmonic, Gabor, and Dihedral frames.

Contribution

It provides a novel construction method for full spark frames using group representations without relying on existing algebraic structures.

Findings

New infinite family of full spark frames constructed

Explicit examples of full spark frames provided

Construction method applicable to various group structures

Abstract

We present a new infinite family of full spark frames in finite dimensions arising from a unitary group representation, where the underlying group is the semi-direct product of a cyclic group by a group of automorphisms. The only previously known algebraically constructed infinite families were the harmonic, Gabor and Dihedral frames. Our construction hinges on a theorem that requires no group structure. Additionally, we illustrate our results by providing explicit constructions of full spark frames.