A note on the quantization for probability measures with respect to the geometric mean error
The author studied the quantization with respect to the geometric mean error for a probability measure \(\mu\) on \(\mathbb{R}^d\) for which there exist some constants \(C,\;\eta >0\) such that \(\mu(B(x,\epsilon))\leq C\epsilon ^{\eta}\) for all \(\epsilon >0\) and all \(x\in \mathbb{R}^d\). For such measure \(\mu\), the author proved that the upper quantization dimension of \(\mu\) is bounded from above by its upper packing dimension and the lower quantization of \(\mu\) is bounded from below by its lower Hausdorff-dimension. This enables us to calculate the quantization dimension for a large class of probability measures which have nice local behavior, including the self-affine measures on general Sierpinski carpets and self-conformal measures. Moreover, based on his previous work, the author proved that the upper and lower quantization coefficient for a self-conformal measure are both positive and finite.
scientific article; zbMATH DE number 6109331
scientific article