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    Best simultaneous approximation in a normed linear space

    Author
    Nairn, Dean C.
    View/Open
    178036_thesis.pdf (21.72Mb)
    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.;
    Metadata
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    URI
    https://hdl.handle.net/20.500.13015/1900
    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.;
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