FRACTAL DIMENSION OF MULTIVARIATE α -FRACTAL FUNCTIONS AND APPROXIMATION ASPECTS

Abstract

In this paper, we explore the concept of dimension preserving approximation of continuous multivariate functions defined on the domain [0, 1]q(= [0, 1] ×⋯ × [0, 1] (q-times) where q is a natural number). We establish a few well-known multivariate constrained approximation results in terms of dimension preserving approximants. In particular, we indicate the construction of multivariate dimension preserving approximants using the concept of α-fractal interpolation functions. We also prove the existence of one-sided approximation of multivariate function using fractal functions. Moreover, we provide an upper bound for the fractal dimension of the graph of the α-fractal function. Further, we study the approximation aspects of α-fractal functions and establish the existence of the Schauder basis consisting of multivariate fractal functions for the space of all real valued continuous functions defined on [0, 1]q and prove the existence of multivariate fractal polynomials for the approximation. © 2022 World Scientific Publishing Company.