A study on Quantization Dimension in complete metric spaces

The primary objective of the present paper is to develop the theory of quantization dimension of an invariant measure associated with an iterated function system consisting of finite number of contractive infinitesimal similitudes in a complete metric space. This generalizes the known results on quantization dimension of self-similar measures in the Euclidean space to a complete metric space. In the last part, continuity of quantization dimension is discussed

Constrained quantization for the Cantor distribution

In this paper, we generalize the notion of unconstrained quantization of the classical Cantor distribution to constrained quantization and give a general definition of constrained quantization. Toward this, we calculate the optimal sets of $n$-points, $n$th constrained quantization errors, the constrained quantization dimensions, and the constrained quantization coefficients taking different families of constraints for all $n\in \mathbb N$. The results in this paper show that both the constrained quantization dimension and the constrained quantization coefficient for the Cantor distribution depend on the underlying constraints. It also shows that the constrained quantization coefficient for the Cantor distribution can exist and be equal to the constrained quantization dimension. These facts are not true in the unconstrained quantization for the Cantor distribution.Comment: arXiv admin note: text overlap with arXiv:2305.1111

Quantization coefficients for uniform distributions on the boundaries of regular polygons

In this paper, we give a general formula to determine the quantization coefficients for uniform distributions defined on the boundaries of different regular $m$-sided polygons inscribed in a circle. The result shows that the quantization coefficient for the uniform distribution on the boundary of a regular $m$-sided polygon inscribed in a circle is an increasing function of $m$, and approaches to the quantization coefficient for the uniform distribution on the circle as $m$ tends to infinity

Constrained quantization for probability distributions

In this paper, for a Borel probability measure $P$ on a Euclidean space $\mathbb R^k$, we extend the definitions of $n$th unconstrained quantization error, unconstrained quantization dimension, and unconstrained quantization coefficient, which traditionally in the literature known as $n$th quantization error, quantization dimension, and quantization coefficient, to the definitions of $n$th constrained quantization error, constrained quantization dimension, and constrained quantization coefficient. The work in this paper extends the theory of quantization and opens a new area of research. In unconstrained quantization, the elements in an optimal set are the conditional expectations in their own Voronoi regions, and it is not true in constrained quantization. In unconstrained quantization, if the support of $P$ contains infinitely many elements, then an optimal set of $n$-means always contains exactly $n$ elements, and it is not true in constrained quantization. It is known that the unconstrained quantization dimension for an absolutely continuous probability measure equals the Euclidean dimension of the underlying space. In this paper, we show that this fact is not true as well for the constrained quantization dimension. It is known that the unconstrained quantization coefficient for an absolutely continuous probability measure exists as a unique finite positive number. From work in this paper, it can be seen that the constrained quantization coefficient for an absolutely continuous probability measure can be any positive number depending on the constraint that occurs in the definition of $n$th constrained quantization error