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Title page for ETD etd-07122013-163741


Type of Document Dissertation
Author Spaeth, Anneliese Heidi
URN etd-07122013-163741
Title A Determination of the Existence of Various Types of Positive Systems in L^p
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Alexander Powell Committee Chair
Akram Aldroubi Committee Member
Alan Peters Committee Member
Doug Hardin Committee Member
Glenn Webb Committee Member
Keywords
  • quasibasis
  • Schauder basis
  • pseudobasis
  • Walsh basis
Date of Defense 2013-04-23
Availability unrestricted
Abstract
We consider various types of generalized bases in spaces of the type L^p(T), where T=[0,1]. More specifically, we determine whether there exists a system {f_n}_n, of the type under consideration, with the property f_n(t)>=0 almost everywhere, for each n in the natural numbers. We refer to a system with the property of almost everywhere non-negativity, as a positive system.

In the spaces with 1<= p < infinity, we determine that there do not exist positive unconditional Schauder bases, and positive unconditional quasibases. In the aforementioned spaces, we determine that there do exist positive conditional quasibases, positive conditional pseudobases, and positive exact systems. In the spaces with 1< p < infinity, we determine that there do not exist positive monotone bases, and that there do exist positive exact systems with exact dual systems. In L^2(T), we demonstrate that there do not exist positive frames, positive orthonormal bases, and positive Riesz bases. Finally, in the spaces with 0< p <= infinity, we show that there do exist positive Hamel bases. Secondary considerations explore product systems on the spaces L^p(T^2).

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