On the box dimension of graph of harmonic functions on the Sierpi\'nski gasket

TL;DR

This paper investigates the fractal dimensions of graphs of harmonic and finite energy functions on the Sierpiński gasket, providing bounds and demonstrating the existence of fractal functions within these classes.

Contribution

It establishes bounds for the box dimension of harmonic and finite energy functions' graphs on the Sierpiński gasket and shows the existence of fractal functions via fractal interpolation.

Findings

Bounds for the box dimension of harmonic functions' graphs.

Bounds for the box dimension of finite energy functions' graphs.

Existence of fractal functions in the energy domain.

Abstract

In this paper, we have obtained bounds for the box dimension of graph of harmonic function on the Sierpi\'nski gasket. Also we get upper and lower bounds for the box dimension of graph of functions that belongs to $\text{dom}(\mathcal{E}),$ that is, all finite energy functionals on the Sierpi\'nski gasket. Further, we show the existence of fractal functions in the function space $\text{dom}(\mathcal{E})$ with the help of fractal interpolation functions. Moreover, we provide bounds for the box dimension of some functions that belong to the family of continuous functions and arise as fractal interpolation functions.