A Characterization of an Element of Best Simultaneous Approximation.
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WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER
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A basic problem of best simultaneous approximation is the following. Given a set S, two or more points not in S, and possibly different measures of the distances from the points to the set, find the element of S which is, in some sense, simultaneously closest to the given points not in S. Deutsch has suggested that some problems of best simultaneous approximation might profitably be viewed as problems of best approximation in an appropriate product space. A few authors have touched upon this approach none, however, have pursued it consistently or developed a completed problem along such a line, even in the simplest of cases. In this paper, we show that Deutschs suggestion can easily be carried out using known results from approximation theory to establish existence, uniqueness, and characterization results. An algorithm guaranteed to converge strongly to the element of best simultaneous approximation under certain circumstances is also proposed. Author
- Theoretical Mathematics