Characterization of strongly non-linear and singular functions by scale space analysis

TL;DR

This paper introduces a novel scale space analysis method to characterize strongly non-linear and singular functions, addressing derivative divergence issues in noiseless physical signals through scale-space velocity operators.

Contribution

It presents a new framework using scale space embedding and velocity operators to analyze non-linear functions, linking to scale relativity and fractional calculus.

Findings

Applied to De Rham's function to demonstrate effectiveness.

Shows how scale space embedding characterizes iterative function system growth.

Provides insights into non-linear signal change analysis.

Abstract

A central notion of physics is the rate of change. While mathematically the concept of derivative represents an idealization of the linear growth, power law types of non-linearities even in noiseless physical signals cause derivative divergence. As a way to characterize change of strongly nonlinear signals, this work introduces the concepts of scale space embedding and scale-space velocity operators. Parallels with the scale relativity theory and fractional calculus are discussed. The approach is exemplified by an application to De Rham's function. It is demonstrated how scale space embedding presents a simple way of characterizing the growth of functions defined by means of iterative function systems.