报告人：朱三国教授 江苏理工学院

报告题目:Asymptotic local uniformity of the quantization error for Ahlfors-David probability measures

报告摘要：Let $\mu$ be an Ahlfors-David probability measures on $\mathbb{R}^q$. For $n\geq 1$, let $\alpha_n$ be an $n$-optimal set for $\mu$ of order $r$ and $(P_a(\alpha_n))_{a\in\alpha_n}$ an arbitrary Voronoi partition with respect to $\alpha_n$. The $n$th quantization error $e_{n,r}(\mu)$ for $\mu$ of order $r$ is given by $e^r_{n,r}(\mu):=\int d(x,\alpha_n)^rd\mu(x)$. Write\[I_a(\alpha,\mu):=\int_{P_a(\alpha_n)}d(x,\alpha_n)^rd\mu(x),\;a\in\alpha_n.\]We prove that, $\underline{J}(\alpha_n,\mu):=\min_{a\in\alpha_n}I_a(\alpha,\mu)$, $\overline{J}(\alpha_n,\mu):=\max_{a\in\alpha_n}I_a(\alpha,\mu)$ and the error difference $e^r_{n,r}(\mu)-e^r_{n+1,r}(\mu)$ are of the same order as $\frac{1}{n}e^r_{n,r}(\mu)$. This, together with Graf and Luschgy's work, yields that all the above three quantities are of the same order as $n^{-(1+\frac{r}{s_0})}$.

报告人简介：朱三国，江苏理工学院教授，德国不来梅大学博士。研究方向是分形测度的离散逼近和量子维数理论。在Math.Z, Proc. AMS等国际知名期刊上发表论文20余篇。

报告时间：2017年12月20日(星期三)下午3：30-4：30

报告地点: 科技楼南楼602室