Fil: Morillas, Patricia Mariela Consejo Nacional de Investigaciones Cientificas y Tecnicas Centro Cientifico Tecnologico Conicet - San Luis Instituto de Matematica Aplicada de San Luis "Prof Ezio Marchi" Universidad Nacional de San Luis Facultad de Ciencias Fisico, Matematicas y Naturales Instituto de Matematica Aplicada de San Luis ; Argentina

/pdf/group-reconstruction-systems-17iv9u8kb6.pdf

Group reconstruction systems

Equiangular tight frames and signature sets in groups

We will present a relation between real equiangular frames and certain special sets in groups which we call signature sets and show that many equiangular frames arise in this manner. Then we will define quasi-signature sets and will examine equiangular frames associated to these subsets of groups. We will extend these results to complex equiangular frames where the inner product between any pair of vectors is a common multiple of a cube root of unity and exhibit equiangular frames that arise from groups in this manner.

Equiangular Frames and Signature Sets

Frames are fundamental tools that are robust to quantization, resilient to additive noise, give stable reconstruction after erasures, and give greater freedom to capture important signal characteristics. Constructing tight frames using various mathematical techniques has been an important area of investigation. In this dissertation we study combinatorial techniques to construct uniform tight frames. We begin by constructing uniform Parseval frames using group representations. We examine conditions on a representation of a group to form a frame representation. In addition, we give an explicit construction of Parseval frame vectors for these frame representations. For insight into applications, we measure the correlation between the frame elements using characters of group representations and give necessary and sufficient conditions for the maximum correlation to be as small as possible. We also derive bounds on the maximum correlation between the frame elements of a tight frame constructed using our techniques. This enables us to differentiate between the behavior of two tight frames in applications. Equiangular tight frames are an important class of finite dimensional frames because of their superior performance and numerous applications. We present a new tool to construct equiangular tight frames using groups and the left regular representation of a group. We prove that many equiangular tight frames arise from subsets of groups which we will call “signature sets”. Subsequently, we define “quasi-signature sets” and examine real equiangular tight frames associated to these subsets of groups. This approach yields further results and establishes new correspondences. We are able to show many examples of equiangular vi tight frames arising from quasi-signature sets. Difference sets are another subsets in groups that are seen to be associated with equiangular frames. We will look at the relationship between difference sets and signature sets. We observe a correspondence between reversible Hadamard difference sets and signature sets. We extend these results to complex equiangular tight frames where the inner product between any pair of vectors is a common multiple of a cube root of unity and exhibit equiangular tight frames that arise from groups in this manner.

/pdf/applications-of-finite-groups-to-parseval-frames-rqkce43ui8.pdf

Applications of finite groups to Parseval frames

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Applications of finite groups to Parseval frames