Dihedral group frames with the Haar property

Abstract

We consider a unitary representation of the Dihedral group obtained by inducing the trivial character from the co-normal subgroup This representation is naturally realized as acting on the vector space We prove that the orbit of almost every vector in with respect to the Lebesgue measure has the Haar property (every subset of cardinality n of the orbit is a basis for ) if n is an odd integer. Moreover, we provide explicit sufficient conditions for vectors in whose orbits have the Haar property. Finally, we derive that the orbit of almost every vector in under the action of the representation has the Haar property if and only if n is odd. This completely settles a problem which was partially solved by the first author.

Keywords:

• Linearly independent frames
• Haar property

AMS Subject Classifications:

• 15A15
• 42C15

Notes

No potential conflict of interest was reported by the authors.