CMX Student/Postdoc Seminar

What global properties of a function can we infer from local data? Remez-type inequalities are one answer to this question: they show the supremum norm of a polynomial can be controlled by its supremum over a small subset of the domain. Remez-type inequalities enjoy widespread use in approximation theory, compressed sensing, and many other areas of applied mathematics. Unfortunately, however, multivariate versions are often limited by a strong dependence on dimension.

We will show in this talk that in fact for many spaces and test sets a dimension-free Remez inequality is available. The proof uses only basic probability and analysis and includes some entertaining ideas, such as how to tensorize a one-dimensional inequality without picking up dimension dependence. Applications to quantum and classical learning theory will also be discussed.

Based on the joint work arXiv:2310.07926 with Lars Becker, Ohad Klein, Alexander Volberg, and Haonan Zhang.