The Marcinkiewicz-type discretization theorems
This work addresses the fundamental problem of norm discretization for finite-dimensional function spaces, which is crucial for applications in approximation theory and numerical analysis.
The paper develops a new technique for discretizing integral norms of functions in finite-dimensional subspaces, combining probabilistic chaining with entropy number results to provide systematic discretization theorems.
The paper is devoted to discretization of integral norms of functions from a given finite dimensional subspace. This problem is very important in applications but there is no systematic study of it. We present here a new technique, which works well for discretization of the integral norm. It is a combination of probabilistic technique, based on chaining, with results on the entropy numbers in the uniform norm.