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Learn about polynomial approximation theory in this 48-minute mathematics seminar that explores Jackson's Inequality on the hypercube of dimension n. Delve into two key findings: the existence of a threshold power n/2 where polynomials of degree less than 0.4999n cannot adequately approximate functions of constant sensitivity, and quantitative error estimates for approximations when the degree approaches n. Discover practical applications including why the sensitivity theorem fails for bounded real-valued functions with approximate degree substitution, and examine a counterexample to the reverse Markov-Bernstein inequality for L1 tail space functions with frequencies of at least 0.4999n. This collaborative research presentation, delivered by Paata Ivanisvili in partnership with Roman Vershynin and Xinyuan Xie, offers valuable insights into advanced mathematical concepts at the intersection of probability theory and functional analysis.
