Dear all, The next Banach spaces webinar is on Friday October 9 9AM CDT (e.g., Dallas, TX time). Please join us at https://unt.zoom.us/j/512907580 Speaker: Vladimir Temlyakov, University of South Carolina Title: Sampling discretization of integral norms Abstract: The talk is devoted to discretization of integral norms of functions from a given finite dimensional subspace. Even though this problem is extremely important in applications, its systematic study has begun recently. In this talk we discuss a conditional theorem for all integral norms $L_q$, $1\le q<\infty$. A new technique, which works well for discretization of the integral norms, was used. It is a combination of probabilistic technique with results on the entropy numbers in the uniform norm. We discuss the behavior of the entropy numbers of classes of functions with bounded integral norms from a given finite dimensional linear subspace. Upper bounds of these entropy numbers in the uniform norm are obtained and applied to establish a Marcinkiewicz type discretization theorem for integral norms of functions from a given finite dimensional subspace. As an application of the general conditional theorem, we discuss a new Marcinkiewicz type discretization for the multivariate trigonometric polynomials with frequencies from the hyperbolic crosses. It is shown that recently developed techniques allow us to improve the known results in this direction. For more information about the past and future talks, and videos please visit the webinar website http://www.math.unt.edu/~bunyamin/banach Thank you, and best regards, Bunyamin Sari