Geodesic
Characterization of the best polynomial approximation with a sign-sensitive weight to a continuous function
Sbornik. Mathematics, Tome 196 (2005) no. 3, pp. 395-422 Cet article a éte moissonné depuis la source Math-Net.Ru

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Necessary and sufficient conditions for the best polynomial approximation with an arbitrary and, generally speaking, unbounded sign-sensitive weight to a continuous function are obtained; the components of the weight can also take infinite values, therefore the conditions obtained cover, in particular, approximation with interpolation at fixed points and one-sided approximation; in the case of the weight with components equal to 1 one arrives at Chebyshev's classical alternation theorem. 
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A. K. Ramazanov. Characterization of the best polynomial approximation with a sign-sensitive weight to a continuous function. Sbornik. Mathematics, Tome 196 (2005) no. 3, pp. 395-422. http://geodesic.mathdoc.fr/item/SM_2005_196_3_a3/

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