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Vanderbilt University

1. Spaeth, Anneliese Heidi. A Determination of the Existence of Various Types of Positive Systems in L^p.

Degree: PhD, Mathematics, 2013, Vanderbilt University

We consider various types of generalized bases in spaces of the type Lp(T), where T=[0,1]. More specifically, we determine whether there exists a system {fn}n, of the type under consideration, with the property fn(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 L2(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 Lp(T2). Advisors/Committee Members: Alexander Powell (chair), Akram Aldroubi (committee member), Doug Hardin (committee member), Alan Peters (committee member), Glenn Webb (committee member).

Subjects/Keywords: quasibasis; Schauder basis; pseudobasis; Walsh basis

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Spaeth, A. H. (2013). A Determination of the Existence of Various Types of Positive Systems in L^p. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://etd.library.vanderbilt.edu/available/etd-07122013-163741/ ;

Chicago Manual of Style (16th Edition):

Spaeth, Anneliese Heidi. “A Determination of the Existence of Various Types of Positive Systems in L^p.” 2013. Doctoral Dissertation, Vanderbilt University. Accessed December 15, 2019. http://etd.library.vanderbilt.edu/available/etd-07122013-163741/ ;.

MLA Handbook (7th Edition):

Spaeth, Anneliese Heidi. “A Determination of the Existence of Various Types of Positive Systems in L^p.” 2013. Web. 15 Dec 2019.

Vancouver:

Spaeth AH. A Determination of the Existence of Various Types of Positive Systems in L^p. [Internet] [Doctoral dissertation]. Vanderbilt University; 2013. [cited 2019 Dec 15]. Available from: http://etd.library.vanderbilt.edu/available/etd-07122013-163741/ ;.

Council of Science Editors:

Spaeth AH. A Determination of the Existence of Various Types of Positive Systems in L^p. [Doctoral Dissertation]. Vanderbilt University; 2013. Available from: http://etd.library.vanderbilt.edu/available/etd-07122013-163741/ ;

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