Graduation Year
2023
Document Type
Dissertation
Degree
Ph.D.
Degree Name
Doctor of Philosophy (Ph.D.)
Degree Granting Department
Mathematics and Statistics
Major Professor
Joel A. Rosenfeld, Ph.D.
Committee Member
Benjamin P. Russo, Ph.D.
Committee Member
Catherine Bénéteau, Ph.D.
Committee Member
Sherwin Kouchekian, Ph.D.
Committee Member
Razvan Teodorescu, Ph.D.
Keywords
Dynamic Mode Decomposition, Koopman, Learning Algorithms, Liouville, Reproducing kernel Hilbert space, Weighted composition operators
Abstract
Modern data science problems revolves around the Koopman operator Cφ (or Composition operator) approach, which provides the best-fit linear approximator to the dynamical system by which the dynamics can be advanced under the discretization. The solution provided by Koopman in the data driven methods is in the sense of strong operator topology, which is nothing better then the point-wise convergence of data (snapshots) in the underlying Hilbert space. Chapter 2 provides the details about the aforementioned issues with essential counter-examples. Thereafter, provable convergence guarantee phenomena is demonstrated by the Liouville weighted composition operators Af,φ over the Fock space by providing the boundedness (cf: Theorem 2.17) and compactness (cf: Theorem 2.25) establishment of Af,φ over the Fock Space. Modern learning algorithms such as support vector machines and activation function for Neural Nets heavily relies on the kernel function which arises from the Lebesgue measure and reproducing kernel of Fock space. Thus, this chapter serves its related results by establishing the norm convergence which benefitsthe practitioners of learning algorithms community.
Chapter 3 investigates the interaction of the weighted composition operators over the Paley-Wiener Space. For sampling and interpolation of signals, Paley-Wiener Space is crucial. The distinctive contribution of this chapter is the discussion of the Phragm ́en-Lindel ̈of indicator function and the P ́olya representation in the presence of the weighted composition operators over the Paley-Wiener Space. Additionally, it has been demonstrated that there exist no compact composition operators over the Paley-Wiener Space. However, due to the presence of an extra symbol (multiplication- ψ) in the weighted composition operator, one can leverage it to attain the compactness of it over the Paley-Wiener Space.
Chapter 4 introduce the Koopman or composition operator over the Poly-logarithmic Hardy Space. As a result of this, we also learned about the Nevanlinna Counting Function for the Poly-logarithmic Hardy Space.
Chapter 5 provides the theory of Mittag-Leffler Space of entire functions in the setting of Lp space. The chief contribution of this chapter are the duality, isometry, boundedness of integral operators via the Schur’s test which leads to perform the atomic decomposition of the Mittag-Leffler Space.
Scholar Commons Citation
Singh, Himanshu, "Applied Analysis for Learning Architectures" (2023). USF Tampa Graduate Theses and Dissertations.
https://digitalcommons.usf.edu/etd/10132
