On Weak Separation Property for Affine Fractal Functions

TL;DR

This paper proves that affine fractal functions lacking the weak separation property are necessarily quadratic functions, revealing a key structural characteristic of such fractal functions.

Contribution

It establishes a novel connection between the weak separation property and the quadratic nature of affine fractal functions.

Findings

Fractal affine functions without weak separation are quadratic.

Weak separation property influences the polynomial degree of fractal functions.

The result characterizes the structure of affine fractal functions based on separation properties.

Abstract

We show that a fractal affine function $f(x)$ defined by a system $\mathcal S$ which does not satisfy weak separation property is a quadratic function.