Geodesic
Best Polynomial Approximations in~$L_2$ of Classes of $2\pi$-Periodic Functions and Exact Values of Their Widths
Matematičeskie zametki, Tome 90 (2011) no. 5, pp. 764-775

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We consider the problem of determining sharp inequalities between the best approximations of periodic differentiable functions by trigonometric polynomials and moduli of continuity of $m$th order in the space $L_2$ as well as present their applications. For some classes of functions defined by these moduli of continuity, we calculate the exact values of $n$-widths in $L_2$.
Keywords: best polynomial approximation, periodic differentiable function, trigonometric polynomial, modulus of continuity, the space $L_2$, $n$-width, Fourier series. 
@article{MZM_2011_90_5_a10,
     author = {M. Sh. Shabozov and G. A. Yusupov},
     title = {Best {Polynomial} {Approximations} in~$L_2$ of {Classes} of $2\pi${-Periodic} {Functions} and {Exact} {Values} of {Their} {Widths}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {764--775},
     publisher = {mathdoc},
     volume = {90},
     number = {5},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2011_90_5_a10/}
}
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M. Sh. Shabozov; G. A. Yusupov. Best Polynomial Approximations in~$L_2$ of Classes of $2\pi$-Periodic Functions and Exact Values of Their Widths. Matematičeskie zametki, Tome 90 (2011) no. 5, pp. 764-775. http://geodesic.mathdoc.fr/item/MZM_2011_90_5_a10/