Asymptotic order of the quantization error for a class of self-similar measures with overlaps
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909570366963712 |
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| author | Zhu, Sanguo |
| author_facet | Zhu, Sanguo |
| contents | Let $\{f_i\}_{i=1}^N$ be a set of equi-contractive similitudes on $\mathbb{R}^1$ satisfying the finite-type condition. We study the asymptotic quantization error for self-similar measures $μ$ associated with $\{f_i\}_{i=1}^N$ and a positive probability vector. With a verifiable assumption, we prove that the upper and lower quantization coefficient for $μ$ are both bounded away from zero and infinity. This can be regarded as an extension of Graf and Luschgy's result on self-similar measures with the open set condition. Our result is applicable to a significant class of self-similar measures with overlaps, including Erdös measure, the $3$-fold convolution of the classical Cantor measure and the self-similar measures on some $λ$-Cantor sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_01103 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic order of the quantization error for a class of self-similar measures with overlaps Zhu, Sanguo Functional Analysis Metric Geometry 28A80, 28A75, 94A15 Let $\{f_i\}_{i=1}^N$ be a set of equi-contractive similitudes on $\mathbb{R}^1$ satisfying the finite-type condition. We study the asymptotic quantization error for self-similar measures $μ$ associated with $\{f_i\}_{i=1}^N$ and a positive probability vector. With a verifiable assumption, we prove that the upper and lower quantization coefficient for $μ$ are both bounded away from zero and infinity. This can be regarded as an extension of Graf and Luschgy's result on self-similar measures with the open set condition. Our result is applicable to a significant class of self-similar measures with overlaps, including Erdös measure, the $3$-fold convolution of the classical Cantor measure and the self-similar measures on some $λ$-Cantor sets. |
| title | Asymptotic order of the quantization error for a class of self-similar measures with overlaps |
| topic | Functional Analysis Metric Geometry 28A80, 28A75, 94A15 |
| url | https://arxiv.org/abs/2403.01103 |