Asymptotic local uniformity of the quantization error for Ahlfors-David probability measures

Abstract

Let $\mu$ be an Ahlfors-David probability measure on ${\mathbb{R}}^q$, namely, there exist some constants $s_0>0$ and ${\epsilon }_0,C_1,C_2>0$ such that \[ C_1ε^{s_0}\leqμ(B(x,ε))\leq C_2ε^{s_0},\;ε\in(0,ε_0),\;x\in{\rm supp}(μ). \] For $n\ge 1$, 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 $e_{n,r}^r(\mu ):=\int d(x,{\alpha }_n{)}^{r}d\mu (x)$. Write \[ I_a(α,μ):=\int_{P_a(α_n)}d(x,α_n)^rdμ(x),\;a\inα_n. \] 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 $e_{n,r}^r(\mu )-e_{n+1,r}^r(\mu )$ are of the same order as $\frac{1}{n}{e}_{n,r}^r(\mu )$. This, together with Graf and Luschgy's work, yields that all the above three quantities are of the same order as $n^{-(1+\frac{r}{s_0})}$.

References in corpus (1)

• The local quantization behavior of absolutely continuous probabilities

Cited by in corpus (1)

• Asymptotic uniformity of the quantization error for Moran measures on ${\mathbb{R}}^1$