A sampling discretization theorem for the square norm of functions from a finite dimensional subspace satisfying Nikol'skii's inequality is proved. The obtained upper bound on the number of sampling points is of the order of the dimension of the subspace.

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This survey addresses sampling discretization and its connections with other areas of mathematics. We present here known results on sampling discretization of both integral norms and the uniform norm beginning with classical results and ending with very recent achievements. We also show how sampling discretization connects to spectral properties and operator norms of submatrices, embedding of finitedimensional subspaces, moments of marginals of high-dimensional distributions, and learning theory. Along with the corresponding results, important techniques for proving those results are discussed as well.

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