
This article is cited in 2 scientific papers (total in 2 papers)
On the best linear method of approximation of some classes analytic functions in the weighted Bergman space
M. S. Saidusaynov^{} ^{} Tajik National University, Dushanbe
Abstract:
In this paper the exact values of different widths in the space
$B_{q,\gamma}$, $1\leq q\leq\infty$ with the weight $\gamma$ for
classes $W_{q,a}^{(r)}(\Phi,\mu)$ were calculated. These classes is
consist from functions $f$, which are analytic in a circle
$U_{R}:=ż: z\leq R\}$ $(0<R\leq 1)$ whose $n
(n\in\mathbb{N})$th derivatives by argument $f_{a}^{(r)}$ is belong
to the space $B_{q,\gamma} (1\leq q\leq\infty, 0<R\leq 1)$ and have
an averaged modulus of smoothness of second order majorized by
function $\Phi$, and everywhere further assumed that the function
$\Phi(t), t>0$ is an arbitrary function that $\Phi(0)=0$.
The exact inequalities between the best polynomial approximation of
analytic functions in a unit disk and integrals consisted from
averaged modulus of smoothness of second order functions with $r$th
derivatives order and concrete weight which is flow out from
substantial meaning of problem statement. The obtained result is
guarantee to calculate the exact values of Bernshtein and
Kolmogorov's widths. Method of approximation which is used for
obtaining the estimation from above the Kolmogorov nwidth is learn
on L. V. Taykov work which earlier is proved for modulus of smoothness
of complex polynomials.
The special interest is offer the problem about constructing the
best linear methods of approximation of classes functions
$W_{q,a}^{(r)}(\Phi,\mu)$ and connected to it the problem in
calculating the exact values of Linear and Gelfand $n$widths. The
founded best linear methods is depend on given number $\mu\geq 1$
and in particular when $\mu=1$ is contain the previous proved
results. Also showed the explicit form an optimal subspaces given
dimension which are implement the values of widths.
Bibliography: 18 titles.
Keywords:
the best linear method, $n$widths, module of smoothness.
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UDC:
517.5 Received: 22.12.2015 Accepted:11.03.2016
Citation:
M. S. Saidusaynov, “On the best linear method of approximation of some classes analytic functions in the weighted Bergman space”, Chebyshevskii Sb., 17:1 (2016), 240–253
Citation in format AMSBIB
\Bibitem{Sai16}
\by M.~S.~Saidusaynov
\paper On the best linear method of approximation of some classes analytic functions in the weighted Bergman space
\jour Chebyshevskii Sb.
\yr 2016
\vol 17
\issue 1
\pages 240253
\mathnet{http://mi.mathnet.ru/cheb467}
\elib{https://elibrary.ru/item.asp?id=25795087}
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This publication is cited in the following articles:

Mukim S. Saidusajnov, “$\mathcal{K}$functionals and exact values of $n$widths in the Bergman space”, Ural Math. J., 3:2 (2017), 74–81

M. Sh. Shabozov, M. S. Saidusainov, “Srednekvadratichnoe priblizhenie funktsii kompleksnoi peremennoi ryadami Fure v vesovom prostranstve Bergmana”, Vladikavk. matem. zhurn., 20:1 (2018), 86–97

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