Fractal Interpolation Functions: A Short Survey

María Antonia Navascués, Arya Kumar Bedabrata Chand, Viswanathan Puthan Veedu, María Victoria Sebastián

Centro Universitario de la Defensa de Zaragoza Academia General Militar, Zaragoza, Spain.
Departamento de Matemática Aplicada EINA, Universidad de Zaragoza, Zaragoza, Spain.
Department of Mathematics, Indian Institute of Technology, Madras, Chennai, India.

DOI: 10.4236/am.2014.512176                

Abstract

The object of this short survey is to revive interest in the technique of fractal interpolation. In order to attract the attention of numerical analysts, or rather scientific community of researchers applying various approximation techniques, the article is interspersed with comparison of fractal interpolation functions and diverse conventional interpolation schemes. There are multitudes of interpolation methods using several families of functions: polynomial, exponential, rational, trigonometric and splines to name a few. But it should be noted that all these conventional nonrecursive methods produce interpolants that are differentiable a number of times except possibly at a finite set of points. One of the goals of the paper is the definition of interpolants which are not smooth, and likely they are nowhere differentiable. They are defined by means of an appropriate iterated function system. Their appearance fills the gap of non-smooth methods in the field of approximation. Another interesting topic is that, if one chooses the elements of the iterated function system suitably, the resulting fractal curve may be close to classical mathematical functions like polynomials, exponentials, etc. The authors review many results obtained in this field so far, although the article does not claim any completeness. Theory as well as applications concerning this new topic published in the last decade are discussed. The one dimensional case is only considered.

Keywords

Fractal Curves, Fractal Functions, Interpolation, Approximation

Conflicts of Interest

The authors declare no conflicts of interest.

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