Geodesic
Best approximations of continuous functions by piecewise monotone functions
Izvestiya. Mathematics , Tome 8 (1974) no. 5, pp. 991-1008.

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In the present article we establish the characteristic theorem of best approximation of a continuous function by piecewise monotone functions. Starting from this theorem we find a sharp estimate for the best approximation. 
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A. S. Loginov. Best approximations of continuous functions by piecewise monotone functions. Izvestiya. Mathematics , Tome 8 (1974) no. 5, pp. 991-1008. http://geodesic.mathdoc.fr/item/IM2_1974_8_5_a2/

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