Dykema, Kenneth2010-01-142010-01-162017-04-072010-01-142010-01-162017-04-072007-052009-05-15http://hdl.handle.net/1969.1/ETD-TAMU-1335Finite 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 ?(E, Fn), where ? is the list of lengths). The family Fk(E, Fn) is shown to be diffeomorphic to the Stiefel manifold Vn(Fk), and Fk ?(E, Fn) is shown to be a smooth manifold if the list of vector lengths ? 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 ?(E, Fn).en-USdifferential geometryharmonic analysisBook