Estimating the diameter of one class of functions in~$L_2$

A. A. Abilov

Matematičeskie zametki, Tome 52 (1992) no. 1, pp. 3-8.

Résumé

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}\|\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 $L_2[-1,1]$.