Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. We pay special attention to the case of trigonometric polynomials with frequencies from an arbitrary finite set with fixed cardinality. We give two different proofs of the fact 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. Previous known results show that one cannot improve on the exponential growth of the number of sampling points for a good discretization theorem in the uniform norm. Also, 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. We illustrate application of our technique on the example of trigonometric polynomials.

翻译：正在研究从一个连续函数的有限维次空间分解功能的统一规范。 我们特别注意从一个固定基点设置的任意限制设置的频率的三角数多元分子的情况。 我们用两种不同的证据来证明,对于连续函数空间的任何一美元维次空间,只要用美元/CN}美元样本点就足以为统一规范使用准确的上限。 先前已知的结果表明,统一规范中良好离散理论取样点数量的指数增长无法改善。 我们还证明了一个一般性结果,将统一规范离散原取样点的上限与与与给定子空间相关的Drichlet内核内核最佳的美元/美元-定期双线近似值联系起来。 我们举例说明了我们技术在三位数多元分子实例中的应用情况。