The Complete Structure of Linear and Nonlinear Deformations of Frames on a Hilbert Space

• Main Author: Agrawal, Devanshu
• Format: Others
• Language: English
• Published: Digital Commons @ East Tennessee State University 2016
• Subjects: Hilbert Space; Reproducing Kernel; Finite Frame; Gabor Frame; Fiber Bundle; Analysis; Mathematics
• Online Access: https://dc.etsu.edu/etd/3003; https://dc.etsu.edu/cgi/viewcontent.cgi?article=4381&context=etd

A frame is a possibly linearly dependent set of vectors in a Hilbert space that facilitates the decomposition and reconstruction of vectors. A Parseval frame is a frame that acts as its own dual frame. A Gabor frame comprises all translations and phase modulations of an appropriate window function. We show that the space of all frames on a Hilbert space indexed by a common measure space can be fibrated into orbits under the action of invertible linear deformations and that any maximal set of unitarily inequivalent Parseval frames is a complete set of representatives of the orbits. We show that all such frames are connected by transformations that are linear in the larger Hilbert space of square-integrable functions on the indexing space. We apply our results to frames on finite-dimensional Hilbert spaces and to the discretization of the Gabor frame with a band-limited window function.