Dihedral group frames with the Haar property. Issue 6 (3rd June 2018)

• Record Type: Journal Article
• Authors: Oussa, Vignon; Sheehan, Brian
• Is Part Of: Linear & multilinear algebra. Volume 66:Issue 6(2018)
• Page Start: 1121
• Page End: 1132
• Publication Date: 2018-06-03
• Subjects: Linearly independent frames -- Haar property; 15A15 -- 42C15; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2017.1337702
• Languages: English
• ISSNs: 0308-1087

Abstract

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.