Iranian Journal of Mathematical Sciences and Informatics
مجله علوم ریاضی و انفورماتیک ایرانیان
IJMSI
Basic Sciences
http://ijmsi.ir
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admin
1735-4463
2008-9473
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8888
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jalali
1388
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gregorian
2009
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online
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fulltext
en
The Basic Theorem and its Consequences
عمومى
General
پژوهشي
Research paper
<p>Let T be a compact Hausdorff topological space and let M denote an n–dimensional subspace of the space C(T ), the space of real–valued continuous functions on T and let the space be equipped with the uniform norm. Zukhovitskii [7] attributes the Basic Theorem to E.Ya.Remez and gives a proof by duality. He also gives a proof due to Shnirel’man, which uses Helly’s Theorem, now the paper obtains a new proof of the Basic Theorem. The significance of the Basic Theorem for us is that it reduces the characterization of a best approximation to f &epsilon C(T ) from M to the case of finite T , that is to the case of approximation in l^{&omega}(r). If one solves the problem for the finite case of T then one can deduce the solution to the general case. An immediate consequence of the Basic Theorem is that for a finite dimensional subspace M of C_{0}(T ) there exists a separating measure forMand f &epsilon C_{0}(T )M, the cardinality of whose support is not greater than dim M+1. This result is a special case of a more general abstract result due to Singer [5]. Then the Basic Theorem is used to obtain a general characterization theorem of a best approximation from M to f &epsilon C(T ). We also use the Basic Theorem to establish the sufficiency of Haar’s condition for a subspace M of C(T ) to be Chebyshev.</p>
Best Approximation, Separating Measure, Chebyshev set.
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http://ijmsi.ir/browse.php?a_code=A-10-1-58&slc_lang=en&sid=1
N. Eftekhari
`10031947532846002099`

10031947532846002099
Yes