Abstract:
In this paper, it is considered the extremal problem of finding the exact constants
in inequalities of Jackson–Stechkin type between the best
approximations of periodic differentiable functions $f\in
L_{2}^{(r)}[0,2\pi]$ by trigonometric polynomials, and the average
values with a positive weight $\varphi$ moduli of continuity of
$m$th order $\omega_{m}(f^{(r)}, t),$ belonging to the space
$L_{p},\, 0<p\le2$. In particular, the problem of
minimizing the constants in these inequalities over all subspaces of
dimension $n,$ raised by N. P. Korneychuk, is solved. For some
classes of functions defined by the specified moduli of continuity, the exact values of $n$-widths of class
\begin{equation*}
L_{2}^{(r)}(m,p,h;\varphi):=\left\{f\in L_{2}^{(r)}: \left(\int\limits_{0}^{h}\omega_{m}^{p}(f^{(r)};t)_{2}\,\varphi(t)dt\right)^{1/p}
\hspace{-1.7mm}\left(\int\limits_{0}^{h}\varphi(t)dt\right)^{-1/p}\le1\right\}
\end{equation*}
are found in the Hilbert space $L_2,$ and the extreme subspace is
identified. In this article, the results are shown which are the
extension and the generalization of some earlier results obtained in
this line of investigation.
Keywords:best approximations, module of continuity of $m$th order, $n$-widths.
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