Author
Nairn, Dean C.
Other Contributors
McLaughlin, H. W.; Habetler, George J.; Ecker, Joseph G.; Kaufman, Howard, 1940-;
Date Issued
1976-12
Subject
Mathematics
Degree
PhD;
Terms of Use
This electronic version is a licensed copy owned by Rensselaer Polytechnic Institute, Troy, NY. Copyright of original work retained by author.;
Abstract
The problem we consider is the approximation of m elements (xₗ,...,xₘ) in a normed linear space by one element in a subset of that normed linear space. To each element in the ordered m-tuple (xₗ,...,xₘ) we associate corresponding continuous seminorms |·|ₗ,...,|·|ₘ. Then using these seminorms to measure the closeness xₒ has to each (xₗ,...,xₘ) we get m errors Eₖ = |xₒ-xₖ|ₖ, k = l,...,m. A best simultaneous approximation to (xₗ,...,xₘ) is the xₒ which minimizes a prescribed combination of the errors Eₗ,...,Eₘ.; Using ideas from mUltiple objective prOgrammi~ we also introduce various generalizations of Bacopoulos' definition of best vectorial approximations. These sets of approximations are shown to have close relations to simultaneous approximations.; The main results of this paper are existence and characterization theorems which generalize well known functional analytica theorems from the theory of best approximation in a normed linear space. When the approximations are from a finite dimensional space we obtain a generalization of Singer's theorem using the extreme elements of the unit ball.; Many authors have considered the above problem in several special cases: approximating a bounded function, finding the restricted Chebyshev center of a finite set, approximating a function with a restricted range, approximating a function with absolute and relative errors, approximating a function and its derivatives, approximating a random function with m possible realizations.;
Description
December 1976; School of Science
Department
Dept. of Mathematical Sciences;
Publisher
Rensselaer Polytechnic Institute, Troy, NY
Relationships
Rensselaer Theses and Dissertations Online Collection;
Access
Restricted to current Rensselaer faculty, staff and students. Access inquiries may be directed to the Rensselaer Libraries.;