Fractal dimension for a class of complex-valued fractal interpolation functions

TL;DR

This paper investigates the fractal dimension of complex-valued fractal functions, comparing it with real-valued cases and establishing bounds, thus advancing understanding of their geometric complexity.

Contribution

It introduces the study of fractal dimensions for complex-valued fractal functions and compares these with real-valued functions, providing new bounds and insights.

Findings

Fractal dimension of complex-valued functions differs from real-valued functions.

Bounds for the fractal dimension of complex-valued fractal functions are established.

Comparison between dimensions of different types of complex and real functions.

Abstract

There are many research papers dealing with fractal dimension of real-valued fractal functions in the recent literature. The main focus of the present paper is to study fractal dimension of complex-valued functions. This paper also highlights the difference between dimensional results of the complex-valued and real-valued fractal functions. In this paper, we study the fractal dimension of the graph of complex-valued function $g(x)+i h(x)$, compare its fractal dimension with the graphs of functions $g(x)+h(x)$ and $(g(x),h(x))$ and also obtain some bounds. Moreover, we study the fractal dimension of the graph of complex-valued fractal interpolation function associated with a germ function $f$, base function $b$ and scaling functions $\alpha_k$.