Optimal Quantization for Uniform Distributions on Cantor-Like Sets

Kreitmeier, Wolfgang

doi:10.1007/s10440-008-9278-3

Cited by 3 publications

(3 citation statements)

References 15 publications

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“…An elementary analysis shows that both sequences n 1/c d ǫ (µ, δ •,n • ) and n 1/c d ǫ (µ −1 , δ •,n • ) are divergent, yet bounded above and below by positive constants, where c = log 2/ log 3 < 1 is the Hausdorff dimension of both the set C = G µ and the measure µ. It seems plausible that Theorem 4.1 can similarly be refined for a wide class of self-similar (singular) distributions, thus complementing known d W -quantization results [18,19,25,29].…”

Section: Best (Unconstrained) Lévy Approximationsmentioning

confidence: 65%

Asymptotics of One-Dimensional Lévy Approximations

Berger

2019

J Theor Probab

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For arbitrary Borel probability measures on the real line, necessary and sufficient conditions are presented that characterize best purely atomic approximations relative to the classical Lévy probability metric, given any number of atoms, and allowing for additional constraints regarding locations or weights of atoms. The precise asymptotics (as the number of atoms goes to infinity) of the approximation error is identified for the important special cases of best uniform (i.e., all atoms having equal weight) and best (unconstrained) approximations, respectively. When compared to similar results known for other probability metrics, the results for Lévy approximations are more complete and require fewer assumptions.Keywords. Best (uniform) approximation, Lévy probability metric, inverse function, inverse measure, approximation error, asymptotic point distribution.MSC2010. 60B10, 60E15, 62E15.

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Section: Best (Unconstrained) Lévy Approximationsmentioning

confidence: 65%

Asymptotics of One-Dimensional Lévy Approximations

Berger

2019

J Theor Probab

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“…[37]) respectively Sierpinski gaskets (cf. [3,14,18]) and by using the results in [22], it should be possible to show, that (20) also holds for these sets, if the contraction factors (c k ) k∈N …”

Section: Open Problems and Concluding Remarksmentioning

confidence: 99%

“…Also for higher-dimensional fractals and the related uniform distributions, the non-existence of the quantization coefficient was shown under special restrictions (cf. [22,30]). is not arithmetic.…”

Section: The Self-similar Casementioning

confidence: 99%

Asymptotic order of quantization for Cantor distributions in terms of Euler characteristic, Hausdorff and Packing measure

Kreitmeier

2008

Journal of Mathematical Analysis and Applications

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For homogeneous one-dimensional Cantor sets, which are not necessarily self-similar, we show under some restrictions that the Euler exponent equals the quantization dimension of the uniform distribution on these Cantor sets. Moreover for a special sub-class of these sets we present a linkage between the Hausdorff and the Packing measure of these sets and the high-rate asymptotics of the quantization error.

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Optimal quantization for nonuniform Cantor distributions

Roychowdhury

2019

Journal of Interdisciplinary Mathematics

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Let P be a Borel probability measure on R such that P = 1 4 P • S −1 1 + 3 4 P • S −1 2 , where S 1 and S 2 are two similarity mappings on R such that S 1 (x) = 1 4 x and S 2 (x) = 1 2 x + 1 2 for all x ∈ R. Such a probability measure P has support the Cantor set generated by S 1 and S 2 . For this probability measure, in this paper, we give an induction formula to determine the optimal sets of n-means and the nth quantization errors for all n ≥ 2. We have shown that the same induction formula also works for the Cantor distribution P := ψ 2 P • S −1 1 + ψ 4 P • S −1 2 supported by the Cantor set generated by S 1 (x) = 1 3 x and S 2 (x) = 1 3 x + 2 3 for all x ∈ R, where ψ is the square root of the Golden ratio 1 2 ( √ 5 − 1). In addition, we give a counter example to show that the induction formula does not work for all Cantor distributions. Using the induction formula we obtain some results and observations which are also given in this paper.2010 Mathematics Subject Classification. 60Exx, 28A80, 94A34.

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Optimal Quantization for Uniform Distributions on Cantor-Like Sets

Cited by 3 publications

References 15 publications

Asymptotics of One-Dimensional Lévy Approximations

Asymptotics of One-Dimensional Lévy Approximations

Asymptotic order of quantization for Cantor distributions in terms of Euler characteristic, Hausdorff and Packing measure

Optimal quantization for nonuniform Cantor distributions

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