Box dimension of generalized affine fractal interpolation functions (II)

TL;DR

This paper investigates the box dimension of generalized affine fractal interpolation functions with vertical scaling, establishing spectral radius properties and providing new dimension estimates, including applications to Weierstrass-type functions.

Contribution

It proves the monotonicity and irreducibility of vertical scaling matrices and uses these to estimate the box dimension of the fractal function's graph.

Findings

Spectral radii of vertical scaling matrices are monotonic.

Under certain conditions, these matrices are irreducible.

The box dimension is estimated via spectral radii limits and sum functions.

Abstract

Let $f$ be a generalized affine fractal interpolation function with vertical scaling functions. In this paper, we prove the monotonicity of spectral radii of vertical scaling matrices without additional assumptions. We also obtain the irreducibility of these matrices under certain conditions. By these results, we estimate $\mathrm{dim}_B \Gamma f$, the box dimension of the graph of $f$, by the limits of spectral radii of vertical scaling matrices. We also estimate $\mathrm{dim}_B \Gamma f$ directly by the sum function of vertical scaling functions. As an application, we study the box dimension of the graph of a generalized Weierstrass-type function.