Date of Award
12-2017
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematical Sciences
Committee Member
Dr. Mishko Mitkovski, Committee Chair
Committee Member
Dr. Taufiquar Khan
Committee Member
Dr. Jeong-Rock Yoon
Committee Member
Dr. Martin Schmoll
Abstract
Suppose H is a separable and complex Hilbert space with a generalized frame (also known as continuous frame) indexed over a metric measure space X. We study the main properties of generalized frames and the operators defined by them, such as concentration operators and Toeplitz operators. Imposing certain localization conditions to the generalized frame, we describe the asymptotic behavior of concentration and Toeplitz operators, and derive important results about the distribution of their eigenvalues. Furthermore, working with multiple generalized frames in H intertwined by a localization conditions, we obtain very general density results. Many examples and applications are shown, among others we obtain necessary density conditions for sampling and interpolation, and these conditions can be applied on classical spaces, such as the Paley-Wiener space, the Bargmann-Fock space, and Gabor systems.
Recommended Citation
Ramírez Flores, Aarón Ernesto, "Localization of Generalized Frames in Hilbert Spaces: Asymptotic Behavior of Concentration and Toeplitz Operators, Sampling and Interpolation, and Density Results" (2017). All Dissertations. 2050.
https://tigerprints.clemson.edu/all_dissertations/2050