2 Data Variation in Coalescence Fractal Interpolation Function

Applied Mathematics, 2014

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.

Journal of Computing in Civil Engineering, 2002

This thesis is devoted to a study about Fractals and Fractal Polynomial Interpolation. Fractal Interpolation is a great topic with many interesting applications, some of which are used in everyday lives such as television, camera, and radio. The thesis is comprised of eight chapters. Chapter one contains a brief introduction and a historical account of fractals. Chapter two is about polynomial interpolation processes such as Newton's, Hermite, and Lagrange. Chapter three focuses on iterated function systems. In this chapter I report results contained in Barnsley's paper, Fractal Functions and Interpolation. I also mention results on iterated function system for fractal polynomial interpolation. Chapters four and five cover fractal polynomial interpolation and fractal interpolation of functions studied by Navascués. Chapter five and six are the generalization of Hermite and Lagrange functions using fractal interpolation. As a concluding chapter we look at the current applications of fractals in various walks of life such as physics and finance and its prospects for the future.

2016

In the present work, the notion of Super Fractal Interpolation Function (SFIF) is introduced for finer simulation of the objects of the nature or outcomes of scientific experiments that reveal one or more structures embedded in to another. In the construction of SFIF, an IFS is chosen from a pool of several IFS at each level of iteration leading to implementation of the desired randomness and variability in fractal interpolation of the given data. Further, an expository description of our investigations on the integral, the smoothness and determination of conditions for existence of derivatives of a SFIF is given in the present work.

Applied Mathematics-A Journal of Chinese Universities, 2021

Fractal interpolation is a modern technique to fit and analyze scientific data. We develop a new class of fractal interpolation functions which converge to a data generating (original) function for any choice of the scaling factors. Consequently, our method offers an alternative to the existing fractal interpolation functions (FIFs). We construct a sequence of α-FIFs using a suitable sequence of iterated function systems (IFSs). Without imposing any condition on the scaling vector, we establish constrained interpolation by using fractal functions. In particular, the constrained interpolation discussed herein includes a method to obtain fractal functions that preserve positivity inherent in the given data. The existence of C r-α-FIFs is investigated. We identify suitable conditions on the associated scaling factors so that α-FIFs preserve r-convexity in addition to the C r-smoothness of original function.

Journal of Mathematical Analysis and Applications, 2007

Based on the construction of Fractal Interpolation Functions, a new construction of Fractal Interpolation Surfaces on arbitrary data is presented and some interesting properties of them are proved. Finally, a lower bound of their box counting dimension is provided.

2012

The real world objects are too irregular to be modeled with the help of traditional interpolation methods. M. F. Barnsley in 1986 proposed the concept of fractal interpolation function (FIF) using iterated function systems (IFS) to describe such real world data. In many cases these data sets represent a curve rather than a function i.e. the data points are not linearly ordered with their abscissa and self affinity is not satisfied in the whole range. The recurrent fractal interpolation function (RFIF) has a role to play in such cases. The purpose of this paper is to apply recurrent fractal interpolation function to fit the piecewise self affine data.

arXiv: Dynamical Systems, 2012

In the present work, the notion of Super Fractal Interpolation Function (SFIF) is introduced for finer simulation of the objects of nature or outcomes of scientific experiments that reveal one or more struc- tures embedded in to another. In the construction of SFIF, an IFS is chosen from a pool of several IFSs at each level of iteration leading to implementation of the desired randomness and variability in fractal interpolation of the given data. Further, an expository description of our investigations on the integral, the smoothness and determination of conditions for existence of derivatives of an SFIF is given in the present work.

SIAM Journal on Mathematical Analysis, 1989

2014

M. F. Barnsley proposed the concept of fractal interpolation function (FIF) using iterated function systems (IFS) to describe the real world objects. The purpose of this paper is to study the parameter identification method for FIF with vertical scaling factor functions (VSFF) for one dimensional data set and establish the generalized version of the analytic approach of Mazel [13].

Academia Letters, 2021