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The Rational Approximation of Real Functions

Daniel E. Wulbert
American Journal of Mathematics
Vol. 100, No. 6 (Dec., 1978), pp. 1281-1315
DOI: 10.2307/2373974
Stable URL: http://www.jstor.org/stable/2373974
Page Count: 35
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The Rational Approximation of Real Functions
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Abstract

Let Rm n(C) be the complex rational functions defined on \lbrack 0,1\rbrack with numerators and denominators having degrees not exceeding m and n respectively. We determine properties of the best uniform approximation from Re Rm n(C)--the real parts of functions in Rm n(C)--to real continuous functions. This article characterizes the closure of Re Rm n(C). For n ≤ m + 1, best approximations are characterized, uniqueness of the approximation is established, and the functions at which the best approximation operator is continuous are identified. The results produce anomalous examples as well as applications to the classical theory of approximation from Rm n(C).

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