A probabilistic approach to discretization inequalities

Abstract: Polynomial discretization inequalities, a central topic in approximation theory, control the norm of a polynomial p on some domain Ω by the norm of p on a finite subset of Ω. The high-dimensional, low-degree regime for p is important in many applications in probability, geometry, and computer science, but has received comparatively little attention. In this talk we introduce a new probabilistic interpolation technique for proving polynomial discretization inequalities in product spaces, yielding dimension-free bounds.

Based on joint work with Lars Becker, Ohad Klein, Alexander Volberg, and Haonan Zhang.