Constrained quantization for the Cantor distribution

TL;DR

This paper extends the theory of constrained quantization to the Cantor distribution, analyzing how constraints affect optimal quantization sets, errors, and dimensions, revealing dependencies not present in unconstrained cases.

Contribution

It generalizes constrained quantization for the Cantor distribution and computes key quantization measures under various constraints, highlighting their dependence on these constraints.

Findings

Constrained quantization dimension depends on the constraints.

Constrained quantization coefficient can exist and equal the dimension.

Results differ from unconstrained quantization for the Cantor distribution.

Abstract

The theory of constrained quantization has been recently introduced by Pandey and Roychowdhury. In this paper, they have further generalized their previous definition of constrained quantization and studied the constrained quantization for the classical Cantor distribution. Toward this, they have calculated 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.