Geodesic
Exact inequalities between the best polynomial approximations and averaged norms of finite differences in the $B_{2}$ space and widths of some classes of functions
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2022), pp. 61-70

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In this paper, exact constants in Jackson–Stechkin type inequalities for characterizing the smoothness of the functions $\Lambda_{m}(f), \ m\in\mathbb{N},$ defined by averaging the norms of finite differences of the $m$-th order of the function $f$ over the argument $z=\rho e^{it}$ analytic in the unit disc belonging $U:=\{z:|z|1\}$ to the Bergman space $B_{2}$ are found. For the classes of analytic functions in the disk $U$, defined by the characteristics of smoothness $\Lambda_{m}(f)$ and $\Phi$ majorants, satisfying a number of conditions, the exact values of various $n$-widths are calculated.
Keywords: generalized modulus of continuity, Jackson–Stechkin type inequality, best approximation, upper boundarie, $n$-widths. 
@article{IVM_2022_3_a6,
     author = {Kh. M. Khuromonov},
     title = {Exact inequalities between the best polynomial approximations and averaged norms of finite differences in the $B_{2}$ space and widths of some classes of functions},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {61--70},
     publisher = {mathdoc},
     number = {3},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2022_3_a6/}
}
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Kh. M. Khuromonov. Exact inequalities between the best polynomial approximations and averaged norms of finite differences in the $B_{2}$ space and widths of some classes of functions. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2022), pp. 61-70. http://geodesic.mathdoc.fr/item/IVM_2022_3_a6/