论文标题
数量估算及其在最佳恢复问题中的应用
Volume estimates and their applications in the problem of optimal recovery
论文作者
论文摘要
我们研究了$ \ Mathbb {r}％ ^{n} $在概率空间上诱导的凸起原点对称体的部分。该方法基于\ john-löwner椭圆形的体积估计以及相应系统引起的规范的期望。获得的估计值使我们能够为部分的半径建立下限，从而为gelfand宽度（或线性跟动）提供了下限。作为应用程序，我们提供了一种新的评估乘数运算符的Gelfand和Kolmogorov宽度的方法。特别是，在困难的情况下，我们建立了标准Sobolev类的尖锐宽度$ W_ {p}^{γ} $，即$ l_ {q} $，即$％1 <q \ leq p \ leq p \ leq p \ leq \ leq \ infty $。
We study volumes of sections of convex origin-symmetric bodies in $\mathbb{R}% ^{n}$ induced by orthonormal systems on probability spaces. The approach is based on volume estimates of \ John-Löwner ellipsoids and expectations of norms induced by the respective systems. The estimates obtained allows us to establish lower bounds for the radii of sections which gives lower bounds for Gelfand widths (or linear cowidths). As an application we offer a new method of evaluation of Gelfand and Kolmogorov widths of multiplier operators. In particular, we establish sharp orders of widths of standard Sobolev classes $W_{p}^{γ}$ in $L_{q}$ in the difficult case, i.e. $% 1<q\leq p\leq \infty $.
