Geodesic
Asymptotic Equalities for Best Approximations for Classes of Infinitely Differentiable Functions Defined by the Modulus of Continuity
Matematičeskie zametki, Tome 99 (2016) no. 6, pp. 904-920

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We obtain asymptotic estimates for best approximations by trigonometric polynomials in the metric of the space $C(L_p)$ for classes of periodic functions expressible as convolutions of kernels $\Psi_\beta$ with Fourier coefficients decreasing to zero faster than any power sequence, and with functions $\varphi\in C$  $(\varphi\in L_p)$ whose moduli of continuity do not exceed the given majorant of $\omega(t)$. It is proved that, in the spaces $C$ and $L_1$, for convex moduli of continuity $\omega(t)$, the obtained estimates are asymptotically sharp.
Keywords: best approximation by trigonometric polynomials, periodic infinitely differentiable function, modulus of continuity, generalized Poisson kernel, linear approximation method, Kolmogorov–Nikol'skii problem. 
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     author = {A. S. Serdyuk and I. V. Sokolenko},
     title = {Asymptotic {Equalities} for {Best} {Approximations} for {Classes} of {Infinitely} {Differentiable} {Functions} {Defined} by the {Modulus} of {Continuity}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {904--920},
     publisher = {mathdoc},
     volume = {99},
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     year = {2016},
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     url = {http://geodesic.mathdoc.fr/item/MZM_2016_99_6_a9/}
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A. S. Serdyuk; I. V. Sokolenko. Asymptotic Equalities for Best Approximations for Classes of Infinitely Differentiable Functions Defined by the Modulus of Continuity. Matematičeskie zametki, Tome 99 (2016) no. 6, pp. 904-920. http://geodesic.mathdoc.fr/item/MZM_2016_99_6_a9/