In this article, we first put forward a new definition of probabilistic Gel’fand-$(N,\delta )$-width which is the classical Gel’fand-N-width in a probabilistic setting. Then we estimate the sharp order of the probabilistic Gel’fand-$(N,\delta )$-width of finite-dimensional space. Furthermore, we obtain the exact order of probabilistic Gel’fand-$(N, \delta )$-width of univariate Sobolev space by the discretization method according to the result of finite-dimensional space.

中文翻译：

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概率环境中的Gel'fand- N-宽度

在本文中，我们首先提出了概率Gel'fand-$（N，\ delta）$-width的新定义，这是概率设置中的经典Gel'fand-N-width。然后我们估计有限维空间的概率Gel'fand-$（N，\ delta）$-宽度的尖锐阶数。此外，根据有限维空间的结果，通过离散化方法获得单变量Sobolev空间的概率Gel'fand-$（N，\ delta）$-宽度的确切顺序。