Geodesic
Mean approximation of functions by Fourier-Gegenbauer sums
Matematičeskie zametki, Tome 3 (1968) no. 5, pp. 587-596.

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Necessary and sufficient conditions for best approximations of functions in the $L^2_{(1-x^2)^\alpha}(-1,1)$ metric, $-1/2\le\alpha1/2$ to zero at a certain rate are established (for $\alpha=?1/2$ known results are obtained). Inequalities for algebraic polynomials are used in the reasoning. 
@article{MZM_1968_3_5_a11,
     author = {S. Z. Rafal'son},
     title = {Mean approximation of functions by {Fourier-Gegenbauer} sums},
     journal = {Matemati\v{c}eskie zametki},
     pages = {587--596},
     publisher = {mathdoc},
     volume = {3},
     number = {5},
     year = {1968},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1968_3_5_a11/}
}
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S. Z. Rafal'son. Mean approximation of functions by Fourier-Gegenbauer sums. Matematičeskie zametki, Tome 3 (1968) no. 5, pp. 587-596. http://geodesic.mathdoc.fr/item/MZM_1968_3_5_a11/