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).
Citation
Strawn, Nathaniel Kirk (2007). Geometry and constructions of finite frames. Master's thesis, Texas A&M University. Available electronically from https : / /hdl .handle .net /1969 .1 /ETD -TAMU -1335.