Multivariate Zipper Fractal Functions

TL;DR

This paper introduces a new multivariate zipper fractal interpolation method, constructing fractal functions with shape-preserving properties and analyzing their approximation capabilities and fractal dimensions.

Contribution

It develops a constructive framework for multivariate zipper fractal functions, including a novel Bernstein zipper fractal, with detailed analysis of their approximation and geometric properties.

Findings

Proposed multivariate zipper fractal functions with shape-preserving features

Derived bounds for the graph of multivariate zipper fractal functions

Analyzed Lipschitz continuity and box dimension of Bernstein zipper fractal functions

Abstract

A novel approach to zipper fractal interpolation theory for functions of several variables is proposed. We develop multivariate zipper fractal functions in a constructive manner. We then perturb a multivariate function to construct its zipper $\alpha$-fractal varieties through free choices of base functions, scaling functions, and a binary matrix called signature. In particular, we propose a multivariate Bernstein zipper fractal function and study its approximation properties such as shape preserving aspects, non-negativity, and coordinate-wise monotonicity. In addition, we derive bounds for the graph of multivariate zipper fractal functions by imposing conditions on the scaling factors and the H\"older exponent of the associated germ function and base function. The Lipschitz continuity of multivariate Bernstein functions is also studied in order to obtain estimates for the box dimension of multivariate Bernstein zipper fractal functions.