Geometry and constructions of finite frames

Abstract

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