Characterization of strongly non-linear and singular functions by scale space analysis. (December 2016)

• Record Type: Journal Article
• Authors: Prodanov, Dimiter
• Is Part Of: Chaos, solitons and fractals. Volume 93(2016)
• Page Start: 14
• Page End: 19
• Publication Date: 2016-12
• Subjects: Fractional calculus -- Singular functions -- Pseudodifferential operators -- Iteration function systems -- Fractals; Primary 26A27: Secondary 26A15 -- 26A33 -- 26A16 -- 47A52 -- 4102; Chaotic behavior in systems -- Periodicals; Solitons -- Periodicals; Fractals -- Periodicals; Chaotic behavior in systems; Fractals; Solitons; Periodicals; 003.7
• DOI: 10.1016/j.chaos.2016.08.010
• Languages: English
• ISSNs: 0960-0779

Abstract

Highlights: Scale-space embeddings and differential operators (scale-space velocities) are introduced. Iteration function systems are characterized by scale-space velocities. The set of growth of the De Rham's function is characterized. Relationship with fractional calculus is demonstrated. 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.