Box dimension of generalized affine fractal interpolation functions

TL;DR

This paper derives explicit formulas for the box dimension of the graph of generalized affine fractal interpolation functions with Lipschitz vertical scaling, using oscillation estimates and vertical scaling matrices.

Contribution

It introduces a method to estimate oscillations and provides explicit formulas for the box dimension under specific conditions, advancing fractal interpolation theory.

Findings

Explicit formula for box dimension of generalized affine fractal functions

Oscillation bounds derived using vertical scaling matrices

Box dimension depends on Lipschitz properties of the scaling function

Abstract

Let $f$ be a generalized affine fractal interpolation function with vertical scaling function $S$. In this paper, we study $\dim_B \Gamma f$, the box dimension of the graph of $f$, under the assumption that $S$ is a Lipschtz function. By introducing vertical scaling matrices, we estimate the upper bound and the lower bound of oscillations of $f$. As a result, we obtain explicit formula of $\dim_B \Gamma f$ under certain constraint conditions.