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：研究从给定有限维连续函数子空间中对函数一致范数的离散化问题。已有结果表明，对于连续函数空间的任意$N$维子空间，需要$e^{CN}$个采样点才能用离散范数精确上界逼近一致范数，且无法在一致范数的良好离散化定理中改进采样点数的指数增长。本文聚焦两类结果，使得能够以$N$的多项式级采样点数实现一致范数的良好离散化。第一种方法通过允许离散范数乘以额外因子（可能依赖于$N$）来放宽离散化不等式；第二种方法对有限维子空间施加限制条件。特别地，我们证明了一个通用结论，该结论将一致范数离散化定理中采样点数的上界与给定子空间所关联的Dirichlet核的最佳$m$项双线性逼近联系起来。