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Locally Piecewise Affine Functions Open Access


Other title
Piecewise Affine Functions
Order Convergence
Functional Analysis
Vector Lattices
Type of item
Degree grantor
University of Alberta
Author or creator
Adeeb, Samer
Supervisor and department
Troitsky, Vladimir (Mathematical Sciences)
Examining committee member and department
Troitsky, Vladimir (Mathematical Sciences)
Tcaciuc, Adi (Mathematical Sciences)
Litvak, Alexander (Mathematical Sciences)
Xanthos, Foivos (Mathematical Sciences)
Kuttler, Jochen (Mathematical Sciences)
Department of Mathematical and Statistical Sciences
Date accepted
Graduation date
Master of Science
Degree level
Piecewise affine functions as defined by Aliprantis and Tourky and denoted by the set S are those functions in C(R^m) that agree with a finite number of affine functions. In this thesis, we extend their study by introducing the set of locally piecewise affine functions denoted by S_lp. Unlike piecewise affine functions, a locally piecewise affine function could possibly agree with an infinite number of affine functions on R^m. We discuss the relationship between the two sets under the umbrella of order theory. In order to define the set of locally piecewise affine functions we first define piecewise affine functions on arbitrary subsets of R^m and discuss the conditions that guarantee the natural extension of a piecewise affine function on arbitrary sets to a piecewise affine function on the whole space. We then define the set of locally piecewise affine functions and discuss how the properties of piecewise affine functions that have been studied previously can be extended to the new set. The literature of vector lattices contains the study of the equivalence or lack thereof of three main definitions for order convergence. However, this problem has not been studied in C(R^m). In this thesis we utilize the results by Anderson and Mathews to study this problem. In doing so, we investigate if C(R^m) possesses the countable sup property which allows us to show that for bounded nets, two main definitions of order convergence in the literature coincide. We also study S and S_lp as sublattices of C(R^m) and we show that both S and subsequently S_lp are order dense minorizing sublattices in C(R^m). We then study the relationship between S and S_lp by introducing the definition of locally finite sets of functions. This definition allows us to show that any locally finite set of functions in S has a supremum and an infimum both of which are in S_lp. In addition, we show that any function in S_lp can be expressed as the difference of the supremums of two locally finite sequences of functions in S. The Stone-Weierstrass theorem can be directly applied to show that piecewise affine functions can uniformly approximate continuous functions on compact sets. However, piecewise affine functions cannot be used to uniformly approximate functions in C(R^m). In this thesis, we show that the set of locally piecewise affine functions can be used to uniformly approximate continuous functions in C(R^m).
Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission.
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