On bivariate fractal approximation

TL;DR

This paper introduces dimension-preserving approximation methods for bivariate functions using fractal interpolation, establishing theoretical results and exploring multi-valued fractal operators for enhanced approximation techniques.

Contribution

It presents new dimension-preserving approximation concepts for bivariate functions and proposes methods based on fractal interpolation functions, expanding the theoretical framework.

Findings

Established results on dimension-preserving approximants

Proposed construction methods using fractal interpolation functions

Defined and studied multi-valued fractal operators

Abstract

In this paper, the notion of dimension preserving approximation for real-valued bivariate continuous functions, defined on a rectangular domain $\rectangle$, has been introduced and several results, similar to well-known results of bivariate constrained approximation in terms of dimension preserving approximants, have been established. Further, some clue for the construction of bivariate dimension preserving approximants, using the concept of fractal interpolation functions, has been added. In the last part, some multi-valued fractal operators associated with bivariate $\alpha$-fractal functions are defined and studied.