Quantitative affine approximation for UMD targets

Abstract

Quantitative affine approximation for UMD targets, Discrete Analysis 2016:6, 48 pp. Let $Y$ be a Banach space. A _martingale difference sequence_ in $Y$ is a sequence of $Y$-valued random variables such that $\mathbb{E}[d_i|d_1,\dots,d_{i-1}]=0$ for every $i$ (and $\mathbb{E}(d_1)=0$). Given $10$, the aim is to find a sub-ball $B'\subset B$ of radius $\rho$, with $\rho$ as large as possible, and an affine function $\Lambda:B'\to Y$, such that $\|f(x)-\Lambda(x)\|\leq\epsilon\rho$ for every $x\in B'$. Importantly, the lower bound on $\rho$ must be independent of the function $f$. The paper obtains a lower bound of $\exp(-(1/\epsilon)^{cn})$ for a constant $c$ that depends on $Y$ only. This improves on the previously best known bound even when $Y$ is a Hilbert space. In the other direction, the best known upper bound (on how large $\rho$ is in the worst case) is singly exponential in $n$, so there is still a large and very interesting gap. There are several other attractive open problems in the paper. The proof of the main result of the paper is not easy, but it contains many new ideas in the geometry of Banach spaces and in harmonic analysis, some of which are interesting results in their own right.