Paata Ivanisvili, UC Irvine

Title: Jackson’s Inequality on the hypercube

Abstract: I will talk about the uniform polynomial approximation problem on the hypercube of dimension n. I will present two results, first indicating that there is a threshold power n/2, i.e., polynomials of degree at most 0.4999n will not always approximate well enough functions of constant sensitivity. The second result, on the opposite side, gives quantitative estimates on the error of approximation when  degree is close to n. There will be two applications presented: one showing that sensitivity theorem does not hold for bounded real valued functions when degree is replaced by approximate degree. The second application will be a counterexample to reverse Markov–Bernstein inequality  for functions in L1 tail space having frequencies at least 0.4999n. This is joint work with Roman Vershynin and Xinyuan Xie.

There will be a projector screen in KAP 414 for those who would like to gather and watch the talk. Note the speaker will not be present in KAP 414.