The quantization for in-homogeneous self-similar measures with in-homogeneous open set condition

Zhu, Sanguo

doi:10.1142/s0129167x15500305

Cited by 3 publications

(3 citation statements)

References 17 publications

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“…Although the supports and mass distributions of the ISMs in the above two cases are completely different (see (2.1) and (2.8)), these ISMs share many properties concerning the asymptotic quantization errors. As is proved in [12,13], for an ISM in Case (i) or (ii), we have D r (µ) = D r (µ) = ξ r := max{ξ 1,r , ξ 2,r };…”

Section: Introductionmentioning

confidence: 61%

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Asymptotics of the geometric mean error in the quantization for in-homogeneous self-similar measures

Zhu

Zhou

Sheng

2016

Journal of Mathematical Analysis and Applications

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We further study the asymptotics of quantization errors for two classes of in-homogeneous self-similar measures µ. We give a new sufficient condition for the upper quantization coefficient for µ to be finite. This, together with our previous work, leads to a necessary and sufficient condition for the upper and lower quantization coefficient of µ to be both positive and finite. Furthermore, we determine (estimate) the convergence order of the quantization error in case that the quantization coefficient is infinite.2000 Mathematics Subject Classification. Primary 28A80, 28A78; Secondary 94A15.

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Section: Introductionmentioning

confidence: 61%

“…By[12, Corollary 1.2], we have ξ 1,r > ξ 2,r for all sufficiently small r > 0. In fact, for any t > 0, we have p 0 ) < 1 (r → 0).…”

mentioning

confidence: 87%

Asymptotics of the geometric mean error in the quantization for in-homogeneous self-similar measures

Zhu

Zhou

Sheng

2016

Journal of Mathematical Analysis and Applications

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“…The L q -spectrum and multifractal properties of such measures have first been treated in [OS08; OS07] and later in the generality needed for our purposes in [Lis14]. This example has already been solved by Zhu in [Zhu15b] for the special choice of a self-similar measure µ, and he was also able to bound the quantization coefficients away from zero and infinity in this context. In the following we will need the function ν : (0, 1) → R given as the solution to N k=1 p q k r ν (q) k = 1.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Quantization dimensions of compactly supported probability measures via Rényi dimensions

Kesseböhmer¹,

Niemann²,

Zhu³

2022

Preprint

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We provide a full picture of the upper quantization dimension in terms of the Rényi dimension, in that we prove that the upper quantization dimension of order r > 0 for an arbitrary compactly supported Borel probability measure ν is given by its Rényi dimension at the point q r where the L q -spectrum of ν and the line through the origin with slope r intersect. In particular, this proves the continuity of r → D r (ν) as conjectured by Lindsay (2001). This viewpoint also sheds new light on the connection of the quantization problem with other concepts from fractal geometry in that we obtain a one-to-one correspondence of the upper quantization dimension and the L q -spectrum restricted to (0, 1). We give sufficient conditions in terms of the L q -spectrum for the existence of the quantization dimension. In this way we show as a byproduct that the quantization dimension exists for every Gibbs measure with respect to a C 1 -self-conformal iterated function system on R d without any assumption on the separation conditions as well as for inhomogeneous self-similar measures under the inhomogeneous open sets condition. Some known general bounds on the quantization dimension in terms of other fractal dimensions can readily be derived from our new approach, some can be improved.

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Quantization dimensions of compactly supported probability measures via Rényi dimensions

Kesseböhmer

Niemann

Zhu

2023

Trans. Amer. Math. Soc.

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We provide a complete picture of the upper quantization dimension in terms of the Rényi dimension by proving that the upper quantization dimension of order r > 0 r>0 for an arbitrary compactly supported Borel probability measure ν \nu is given by its Rényi dimension at the point q r q_{r} where the L q L^{q} -spectrum of ν \nu and the line through the origin with slope r r intersect. In particular, this proves the continuity of r ↦ D ¯ r ( \nu ) r\mapsto \overline {D}_{r}(\text {\nu )} as conjectured by Lindsay [Quantization dimension for probability distributions, ProQuest LLC, Ann Arbor, MI, 2001. Thesis (Ph.D.)-University of North Texas]. This viewpoint also sheds new light on the connection of the quantization problem with other concepts from fractal geometry in that we obtain a one-to-one correspondence of the upper quantization dimension and the L q L^{q} -spectrum restricted to ( 0 , 1 ) \left (0,1\right ) . We give sufficient conditions in terms of the L q L^{q} -spectrum for the existence of the quantization dimension. In this way we show as a byproduct that the quantization dimension exists for every Gibbs measure with respect to a C 1 \mathcal {C}^{1} -self-conformal iterated function system on R d \mathbb {R}^{d} without any assumption on the separation conditions as well as for inhomogeneous self-similar measures under the inhomogeneous open sets condition. Some known general bounds on the quantization dimension in terms of other fractal dimensions can readily be derived from our new approach, some can be improved.

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The quantization for in-homogeneous self-similar measures with in-homogeneous open set condition

Cited by 3 publications

References 17 publications

Asymptotics of the geometric mean error in the quantization for in-homogeneous self-similar measures

Asymptotics of the geometric mean error in the quantization for in-homogeneous self-similar measures

Quantization dimensions of compactly supported probability measures via Rényi dimensions

Quantization dimensions of compactly supported probability measures via Rényi dimensions

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