Geometry and constructions of finite frames

Abstract

Finite frames are special collections of vectors utilized in Harmonic Analysis and Digital Signal Processing. In this thesis, geometric aspects and construction techniques are considered for the family of k-vector frames in Fn = Rn or Cn sharing a fixed frame operator (denoted Fk(E, Fn), where E is the Hermitian positive definite frame operator), and also the subfamily of this family obtained by fixing a list of vector lengths (denoted Fk µ(E, Fn), where µ is the list of lengths). The family Fk(E, Fn) is shown to be diffeomorphic to the Stiefel manifold Vn(Fk), and Fk µ(E, Fn) is shown to be a smooth manifold if the list of vector lengths µ satisfy certain conditions. Calculations for the dimensions of these manifolds are also performed. Finally, a new construction technique is detailed for frames in Fk(E, Fn) and Fk µ(E, Fn).