Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. Previous known results show that for any $N$-dimensional subspace of the space of continuous functions it is sufficient to use $e^{CN}$ sample points for an accurate upper bound for the uniform norm by the discrete norm and that one cannot improve on the exponential growth of the number of sampling points for a good discretization theorem in the uniform norm. In this paper we focus on two types of results, which allow us to obtain good discretization of the uniform norm with polynomial in $N$ number of points. In the first way we weaken the discretization inequality by allowing a bound of the uniform norm by the discrete norm multiplied by an extra factor, which may depend on $N$. In the second way we impose restrictions on the finite dimensional subspace under consideration. In particular, we prove a general result, which connects the upper bound on the number of sampling points in the discretization theorem for the uniform norm with the best $m$-term bilinear approximation of the Dirichlet kernel associated with the given subspace.

翻译：本文研究连续函数空间中给定有限维子空间上函数一致范数的离散化问题。已有结果表明，对连续函数空间的任意N维子空间，需使用e^{CN}个采样点才能通过离散范数精确上界估计一致范数，且无法在一致范数最优离散化定理中改进采样点数的指数增长特征。本文聚焦两类可显著减少采样点数的研究成果，其采样点数呈N的多项式增长。第一种方法通过允许离散范数乘以与N相关的额外因子来弱化离散化不等式；第二种方法则对有限维子空间施加约束条件。特别地，我们证明了一个重要结论：该结论将一致范数离散化定理中采样点数的上界估计，与被研究子空间对应的Dirichlet核的最佳m项双线性逼近直接相关联。