Sampling discretization and related problems

Abstract

This survey addresses sampling discretization and its connections with other areas of mathematics. The survey concentrates on sampling discretization of norms of elements of finite-dimensional subspaces. 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 finite-dimensional 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.

References in corpus (10)

• Integral norm discretization and related problems
• On Nikol'skii inequalities for domains in $R^d$
• A note on sampling recovery of multivariate functions in the uniform norm
• The Marcinkiewicz-type discretization theorems for the hyperbolic cross polynomials
• Entropy numbers and Marcinkiewicz-type discretization theorem
• On optimal recovery in $L_2$
• Function values are enough for $L_2$-approximation: Part II
• Bounds on Kolmogorov widths and sampling recovery for classes with small mixed smoothness
• Sampling discretization error for function classes
• A Bernstein type inequality

Cited by in corpus (1)

• Marcinkiewicz-Zygmund Inequalities for Polynomials in Fock Space