arXiv subjects

Tanmoy Som

Publications and source records attributed to Tanmoy Som.

5 recordsLinked to original sources

arXiv2026

Generalized multivariate Fractal Interpolation Function and $\alpha$-Fractal Function

In this paper, we introduce a new approach for constructing multivariate fractal interpolation functions and $\alpha$-fractal functions associated with multivariate functions. Unlike the existing methods that rely on the Banach contraction principle, here the construction is based on Matkowski and Rakotch contractions. While numerous methods for constructing multivariate fractal interpolation functions have been explored in the literature, the approach given in this paper is distinct in the sense that it generalizes all previously known techniques and provides a broader framework for such constructions. We propose a technique to develop nonlinear iterated function systems using the generalized contractions and establish that the attractors of such systems are the graphs of continuous multivariate functions interpolating theoretical data points. Furthermore, for the Rakotch contractions, the existence of an invariant Borel probability measure supported on the graph of the associated multivariate fractal interpolation function is explored.

math.FA↗

arXiv2023

Some results on Continuous dependence of fractal functions on the Sierpiński gasket

In this article, we show that $α$-fractal functions defined on Sierpiński gasket (denoted by $\triangle$) depend continuously on the parameters involved in the construction. In the latter part of this article, the continuous dependence of parameters on $α$-fractal functions defined on $\triangle$ is shown graphically.

math.FA↗

arXiv2023

Generalized Hausdorff metric on $S_{b}$-metric space and some fixed point results

In this paper, a metric on $S_b$-metric space analogous to the Hausdorff metric has been introduced and some basic properties are obtained on multi-valued $S_b$-metric space. Further, the fundamental multi-valued contraction of Nadler(1962) has been extended to the $S_b$-metric space setting, and two results have been established. The entire study is supported by suitable examples.

math.FA↗

arXiv2022

Set-valued α-fractal functions

In this paper, we introduce the concept of the $α$-fractal function and fractal approximation for a set-valued continuous map defined on a closed and bounded interval of real numbers. Also, we study some properties of such fractal functions. Further, we estimate the perturbation error between the given continuous function and its $α$-fractal function. Additionally, we define a new graph of a set-valued function different from the standard graph introduced in the literature and establish some bounds on the fractal dimension of the newly defined graph of some special classes of set-valued functions. Also, we explain the need to define this new graph with examples. In the sequel, we prove that this new graph of an $α$-fractal function is an attractor of an iterated function system.

math.FA↗

arXiv2022

Box Dimension and Fractional Integrals of Multivariate Fractal Interpolation Functions

In this article, we construct the multivariate fractal interpolation functions for a given data points and explore the existence of $α$-fractal function corresponding to the multivariate continuous function defined on $[0,1]\times \cdots \times [0,1](q\text{-times})$. The parameters are selected such that the corresponding fractal version preserves some of the original function's properties, for instance, if the given function is Hölder continuous, then the corresponding $α$-fractal function is also Hölder continuous. Moreover, we explore the restriction of the $α$-fractal function on the co-ordinate axis. Furthermore, the box dimension and Hausdorff dimension of the graph of the multivariate $α$-fractal function and its restriction are investigated. In the last section, we prove that the mixed Riemann-Liouville fractional integral of fractal function satisfies a self-referential equation.

math.FA↗