Geodesic
The simultaneous interpolation and one-sided approximation in the mean of continuous functions
Matematičeskie zametki, Tome 12 (1972) no. 6, pp. 701-712.

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We consider the best one-sided approximation in the mean of a continuous function with simultaneous interpolation of this function by Chebyshev polynomials. It is proved that the polynomial of best approximation for this problem is unique in the case when the Chebyshev system is differentiable three times and the function being approximated has a continuous third-order derivative. It is shown that the conditions imposed are essential. 
@article{MZM_1972_12_6_a6,
     author = {V. A. Shmatkov},
     title = {The simultaneous interpolation and one-sided approximation in the mean of continuous functions},
     journal = {Matemati\v{c}eskie zametki},
     pages = {701--712},
     publisher = {mathdoc},
     volume = {12},
     number = {6},
     year = {1972},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1972_12_6_a6/}
}
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V. A. Shmatkov. The simultaneous interpolation and one-sided approximation in the mean of continuous functions. Matematičeskie zametki, Tome 12 (1972) no. 6, pp. 701-712. http://geodesic.mathdoc.fr/item/MZM_1972_12_6_a6/