THE BASIC THEOREM AND ITS CONSEQUENCES

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EFTEKHARI N.

مرکز اطلاعات علمی Scientific Information Database (SID) - Trusted Source for Research and Academic Resources

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Journal: Iranian Journal of Mathematical Sciences and Informatics Year:2009 | Volume:4 | Issue:1 Page(s): 27-35

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Title

THE BASIC THEOREM AND ITS CONSEQUENCES

Author(s)

EFTEKHARI N. | Issue Writer Certificate

Keywords

BEST APPROXIMATION

SEPARATING MEASURE

CHEBYSHEV SET

Abstract

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 Î C(T ) from M to the case of finite T , that is to the case of approximation in l¥(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 C0(T ) there exists a SEPARATING MEASURE for M and f Î C0(T)\M, the cardinality of whose support is not greater than dimM+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 Î 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.

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APA: Copy

EFTEKHARI, N.. (2009). THE BASIC THEOREM AND ITS CONSEQUENCES. IRANIAN JOURNAL OF MATHEMATICAL SCIENCES AND INFORMATICS (IJMSI), 4(1), 27-35. SID. https://sid.ir/paper/310278/en

Vancouver: Copy

EFTEKHARI N.. THE BASIC THEOREM AND ITS CONSEQUENCES. IRANIAN JOURNAL OF MATHEMATICAL SCIENCES AND INFORMATICS (IJMSI)[Internet]. 2009;4(1):27-35. Available from: https://sid.ir/paper/310278/en

IEEE: Copy

N. EFTEKHARI, “THE BASIC THEOREM AND ITS CONSEQUENCES,” IRANIAN JOURNAL OF MATHEMATICAL SCIENCES AND INFORMATICS (IJMSI), vol. 4, no. 1, pp. 27–35, 2009, [Online]. Available: https://sid.ir/paper/310278/en

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