|
This article is cited in 4 scientific papers (total in 4 papers)
Estimating the diameter of one class of functions in $L_2$
A. A. Abilov Daghestan State University
Abstract:
Let
\begin{gather*}
f(x)\in L_2[-1,1], \quad \|f\|=\sqrt{\int^1_{-1}|f(x)|^z\,dx},
\\
f_h(x)=\frac1\pi\int_0^{\pi}f(x\cos h+\sqrt{1-x^2}\sin h\cos\pi)\,d\theta, \quad h>0,
\\
\widetilde{\omega}(f^{(r)},t)=\sup_{0<h<t}\|\sqrt{(1-x^2)^r}[f^{(r)}(x)-f_h^{(r)}(x)]\|,
\\
\widetilde{W}_{\omega}^r=\{f\in L_2[-1,1]:\widetilde{\omega}(f^{(r)};t)\leqslant c\omega(t)\},
\end{gather*}
where $r=0,1,2,\dots,\omega(t)$ is a given modulus of continuity, and $c>0$ is a constant. The estimate is piroved, where $d_n(\widetilde{W}_{\omega}^r;L_2[-1,1])\asymp n^{-r}\omega(n^{-r})$ ($n>r$) is the Kolmogorov $n$-diameter of the set $\widetilde{W}_{\omega}^r$ in the space $L_2[-1,1]$.
Received: 18.08.1989
Citation:
A. A. Abilov, “Estimating the diameter of one class of functions in $L_2$”, Mat. Zametki, 52:1 (1992), 3–8; Math. Notes, 52:1 (1992), 631–635
Linking options:
https://www.mathnet.ru/eng/mzm4647 https://www.mathnet.ru/eng/mzm/v52/i1/p3
|
Statistics & downloads: |
Abstract page: | 202 | Full-text PDF : | 95 | First page: | 1 |
|