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

Abstract

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.

Analytical and Numerical Treatments of Conservative Diffusions and the Burgers Equation

“…Under this assumption, we have Product rule Quotient rule where wherever . For compositions of functions, and and Basic evaluation formula [ 51 ]: Derivative regularization [ 52 ]: Let be composition with , a -differentiable function at x , then where is the fractal q -adic (co-)variation.…”

Generalized Differentiability of Continuous Functions

“…Such an opposition leads to the question whether the ordinary (i.e., Cauchy) derivatives can be generalized locally in a way that can describe such irregular objects. This is certainly achievable for a class of singular functions, such as the De Rham's function, which can be characterized locally in terms of fractional velocity [4,5]. Purely mathematically, the derivatives can be generalized in several ways.…”

Fractional Velocity as a Tool for the Study of Non-Linear Problems

“…Mathematical descriptions of strongly non-linear phenomena necessitate relaxation of the assumption of differentiability [26]. While this can be achieved also by fractional differintegrals, or by multiscale approaches [7], the present work focuses on local descriptions in terms of limits of difference quotients [6] and nonlinear scale-space transformations [30]. The reason for this choice is that locality provides a direct way of physical interpretation of the obtained results.…”

“…The proof is given in [30] and will not be repeated. In this formulation the value of 1 − β can be considered as the magnitude of deviation from linearity (or differentiability) of the signal at that point.…”