UAlberta-PIMS Mathematics and Statistics Colloquium: Egor Kosov

Informally speaking, sampling discretization studies how well one can replace the computa- tion of integral L p norms for a given class of functions by the evaluation of these functions at a fixed small set of points. On the one hand, such problems are classical. The first results of this type were obtained in 1937 by Marcinkiewicz for L p -norms with p ∈ (1, +∞) and by Marcinkiewicz–Zygmund for L 1 -norm for the class of all univariate trigonometric polynomials of a fixed degree. On the other hand, the systematic study of sampling discretization has begun only recently. In more detail, let C(Ω) be the space of all continuous functions on some compact subset Ω of R n equipped with a probability Borel measure μ. Let XN be some N-dimensional subspace of C(Ω), let p ∈ [1, +∞), and ε ∈ (0, 1). We aim to determine the least possible integer m such that there are points x1, . . . , , xm ∈ Ω, for which (1 − ε)∥f∥ p p ≤ 1 m Xm j=1 |f(xj )| p ≤ (1 + ε)∥f∥ p p ∀f ∈ XN where ∥f∥ p p := R Ω |f(x)| pμ(dx) and ∥f∥∞ := max{|f(x)|: x ∈ Ω}. Clearly, m can’t be less than the dimension N of the subspace XN . Thus, we are interested in the conditions on the subspace which can guarantee that the number of points m can be chosen close to the dimension N. In the talk we are going to discuss the recent progress and some techniques in this area.