arXiv subjects

Md. N. Akhtar

Publications and source records attributed to Md. N. Akhtar.

2 recordsLinked to original sources

arXiv2024

Quantization of Cantor-Like Set on the Real Projective Line

In this article, an iterated function system (IFS) is considered on the real projective line $\mathbb{RP}^1$ so that the attractor is a Cantor-like set. Hausdorff dimension of this attractor is estimated. The existence of a probability measure associated with this IFS on $\mathbb{RP}^1$ is also demonstrated. It is shown that the $n$-th quantization error of order $r$ for the push-forward measure is a constant multiple of the $n$-th quantization error of order $r$ of the original measure. Finally, an upper bound for the $n$-th quantization error of order $2$ for this measure is provided.

math.DS↗

arXiv2023

Fractal Interpolation on the Real Projective Plane

Formerly the geometry was based on shapes, but since the last centuries this founding mathematical science deals with transformations, projections and mappings. Projective geometry identifies a line with a single point, like the perspective on the horizon line and, due to this fact, it requires a restructuring of the real mathematical and numerical analysis. In particular, the problem of interpolating data must be refocused. In this paper we define a linear structure along with a metric on a projective space, and prove that the space thus constructed is complete. Then we consider an iterated function system giving rise to a fractal interpolation function of a set of data.

math.DS↗