The paper addresses a problem of sampling discretization of integral norms of elements of finite-dimensional subspaces satisfying some conditions. We prove sampling discretization results under a standard assumption formulated in terms of the Nikol'skii-type inequality. {In particular, we obtain} some upper bounds on the number of sample points sufficient for good discretization of the integral $L_p$ norms, $1\le p<2$, of functions from finite-dimensional subspaces of continuous functions. Our new results improve upon the known results in this direction. We use a new technique based on deep results of Talagrand from functional analysis.

翻译：本文研究了满足某些条件的有限维子空间元素积分范数的采样离散化问题。我们在基于尼科尔斯基型不等式的标准假设下证明了采样离散化结果。特别地，我们获得了在连续函数有限维子空间中，足以对$L_p$范数（$1\le p<2$）进行良好离散化的采样点数的若干上界。我们的新结果改进了该方向的已知结论。我们采用了一种基于塔拉格兰德泛函分析深层次结果的新技术。