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 $L^1$ tail space having frequencies at least $0.4999n$. This is joint work with Roman Vershynin and Xinyuan Xie.

analysis of PDEsclassical analysis and ODEsfunctional analysis

Audience: researchers in the topic

OARS Online Analysis Research Seminar

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