Geodesic
Approximation of functions by polynomials with Hermitian interpolation and restrictions on the coefficients
Izvestiya. Mathematics , Tome 67 (2003) no. 1, pp. 183-206

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We determine the order of approximation of real functions of finite smoothness by polynomials with various restrictions: approximation of functions and their derivatives with Hermitian interpolation at the endpoints, one-sided approximation (for example, from above), and assumptions on the coefficients. 
@article{IM2_2003_67_1_a9,
     author = {R. M. Trigub},
     title = {Approximation of functions by polynomials with {Hermitian} interpolation and restrictions on the coefficients},
     journal = {Izvestiya. Mathematics },
     pages = {183--206},
     publisher = {mathdoc},
     volume = {67},
     number = {1},
     year = {2003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2003_67_1_a9/}
}
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R. M. Trigub. Approximation of functions by polynomials with Hermitian interpolation and restrictions on the coefficients. Izvestiya. Mathematics , Tome 67 (2003) no. 1, pp. 183-206. http://geodesic.mathdoc.fr/item/IM2_2003_67_1_a9/