TITLE:
Fractal Interpolation Functions: A Short Survey
AUTHORS:
María Antonia Navascués, Arya Kumar Bedabrata Chand, Viswanathan Puthan Veedu, María Victoria Sebastián
KEYWORDS:
Fractal Curves, Fractal Functions, Interpolation, Approximation
JOURNAL NAME:
Applied Mathematics,
Vol.5 No.12,
June
26,
2014
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.