A. K. B. Chand

Monotonicity Preserving Rational Quadratic Fractal Interpolation Functions

Fractal interpolation is an advanced technique for analysis and synthesis of scientific and engineering data. We introduce the -rational quadratic fractal interpolation functions (FIFs) through a suitable rational quadratic iterated function system (IFS). The novel notion of shape preserving fractal interpolation without any shape parameter is introduced through the rational fractal interpolation model in the literature for the first time. For a prescribed set of monotonic data, we derive the sufficient conditions by restricting the scaling factors for shape preserving -rational quadratic FIFs. A local modification pertaining to any subinterval is possible in this model if the scaling factors are chosen appropriately. We establish the convergence results of a monotonic rational quadratic FIF to the original function in . For given data with derivatives at grids, our approach generates several monotonicity preserving rational quadratic FIFs, whereas this flexibility is not available in the classical approach. Finally, numerical experiments support the importance of the developed rational quadratic IFS scheme through construction of visually pleasing monotonic rational fractal curves including the classical one.

Approximation using hidden variable fractal interpolation function

The notion of hidden variable fractal interpolation provides a method to approximate functions that are self-referential or non-self-referential, and consequently allows great flexibility and diversity for the fractal modeling problem. The current article intends to apply hidden variable fractal interpolation to associate a class of R-valued continuous fractal functions with a prescribed continuous function. Suitable values of the parameters are identified so that the fractal functions retain positivity and regularity of the germ function. As an application of the developed theory, we obtain positive C-cubic spline hidden variable fractal interpolation functions corresponding to a prescribed set of positive data, thus initiating a new approach to shape preserving approximation via hidden variable fractal function. Depending on the values of the parameters, these positive interpolants can reflect the self-referentiality or non-self-referentiality of the original data defining function and fractality of its derivative. Therefore, the present scheme outperforms the traditional nonrecursive positivity preserving C-cubic spline interpolation scheme and its fractal extension studied recently in the literature. Mathematics Subject Classification (2010). Primary: 28A80; Secondary: 41A05, 41A10, 41A29, 41A30.

Fractal Polynomials and Maps in Approximation of Continuous Functions

ABSTRACT The primary goal of this article is to establish some approximation properties of fractal functions. More specifically, we establish that a monotone continuous real-valued function can be uniformly approximated with a monotone fractal polynomial, which in addition agrees with the function on an arbitrarily given finite set of points. Furthermore, the simultaneous approximation and \mboxinterpolation which is norm-preserving property of fractal polynomials is established. In the final part of the article, we establish differentiability of a more general class of fractal functions. It is shown that these smooth fractal functions and their derivatives are good approximants for the original function and its \mboxderivatives.

A New Class of Rational Quadratic Fractal Functions with Positive Shape Preservation

Fractal interpolation functions (FIFs) developed through iterated function systems prove more general than their classical counterparts. However, the theory of fractal interpolation functions in the domain of shape preserving interpolation is not fully explored. In this paper, we introduce a new kind of iterated function system (IFS) involving rational functions of the form \(\frac{p_{n}(x)} {q_{n}(x)}\), where p n (x) are quadratic polynomials determined through the interpolation conditions of the corresponding FIF and q n (x) are preassigned quadratic polynomials involving one free shape parameter. The presence of the scaling factors in our rational FIF adds a layer of flexibility to its classical counterpart and provides fractality in the derivative of the interpolant. The uniform convergence of the rational quadratic FIF to the original data generating function is established. Suitable conditions on the rational IFS parameters are developed so that the corresponding rational quadratic fractal interpolant inherits the positivity property of the given data.

Constrained Data Visualization Using Rational Bi-cubic Fractal Functions

This paper addresses a method to obtain rational cubic fractal functions, which generate surfaces that lie above a plane via blending functions. In particular, the constrained bivariate interpolation discussed herein includes a method to construct fractal interpolation surfaces that preserve positivity inherent in a prescribed data set. The scaling factors and shape parameters involved in fractal boundary curves are constrained suitably such that these fractal boundary curves are above the plane whenever the given interpolation data along the grid lines are above the plane. Our rational cubic spline FIS is above the plane whenever the corresponding fractal boundary curves are above the plane. We illustrate our interpolation scheme with some numerical examples.

Construction of Fractal Bases for Spaces of Functions

The construction of fractal versions of classical functions as polynomials, trigonometric maps, etc. by means of a particular Iterated Function System of the plane is tackled. The closeness between the classical function and its fractal analogue provides good properties of approximation and interpolation to the latter. This type of methodology opens the use of non-smooth and fractal functions in approximation. The procedure involves the definition of an operator mapping standard functions into their dual fractals. The transformation is linear and bounded and some bounds of its norm are established. Through this operator we define families of fractal functions that generalize the classical Schauder systems of Banach spaces and the orthonormal bases of Hilbert spaces. With an appropriate election of the coefficients of Iterated Function System we define sets of fractal maps that span the most important spaces of functions as \({\mathcal {C}}[a,b]\) or \({\mathcal {L}}^p[a,b]\).

Quintic Hermite Fractal Interpolation in a Strip: Preserving Copositivity

The notion of fractal interpolation provides a general framework which includes traditional nonrecursive splines as special cases. In this paper, we describe a procedure for the construction of quintic Hermite FIFs as \(\alpha \)-fractal function corresponding to the classical quintic Hermite interpolant. In contrast to traditional piecewise nonrecursive quintic Hermite interpolant, its fractal version has a second derivative which is differentiable in a finite or dense subset of the interpolation interval. This scheme offers an additional freedom over the classical quintic Hermite interpolants due to the presence of scaling factors. The elements of the iterated function system are identified so that the class of \(\alpha \)-fractal function \(f^\mathbf {\alpha }\) reflects the fundamental shape properties such as positivity, monotonicity, and convexity in addition to the regularity of f in the given interval. Using this general theory, an algorithm for positivity of quintic Hermite FIF is presented. Finally, the algorithm for a quintic Hermite fractal interpolants copositive with a given data set is prescribed.

A Novel Approach to Surface Interpolation: Marriage of Coons Technique and Univariate Fractal Functions

The current article is intended to demonstrate that the theory of fractal functions when applied in conjunction with methods in the classical numerical analysis can supply new solution techniques that supplement and subsume the existing ones. To this end, in the first part of the paper, we review a \(\mathscr {C}^1\)-continuous rational cubic fractal interpolation function (FIF) introduced recently [Viswanathan and Chand, Elec. Trans. Numer. Anal. 41 (2014), pp. 420–442]. We carry out the convergence analysis of this univariate rational FIF and determine suitable values of the derivative parameters so that its global smoothness enhances to \(\mathscr {C}^2\). In the subsequent part of the article, we apply Coons technique of transfinite interpolation in order to construct a new kind of \(\mathscr {C}^1\)-continuous bivariate fractal interpolation surface.

Toward a Unified Methodology for Fractal Extension of Various Shape Preserving Spline Interpolants

Fractal interpolation, one in the long tradition of those involving the interpolatary theory of functions, is concerned with interpolation of a data set with a function whose graph is a fractal or a self-referential set. The novelty of fractal interpolants lies in their ability to model a data set with either a smooth or a nonsmooth function depending on the problem at hand. To broaden their horizons, some special class of fractal interpolants are introduced and their shape preserving aspects are investigated recently in the literature. In the current article, we provide a unified approach for the fractal generalization of various traditional nonrecursive polynomial and rational splines. To this end, first we shall view polynomial/rational FIFs as \(\alpha \)-fractal functions corresponding to the traditional nonrecursive splines. The elements of the iterated function system are identified befittingly so that the class of \(\alpha \)-fractal function \(f^\mathbf {\alpha }\) incorporates the geometric features such as positivity, monotonicity, and convexity in addition to the regularity inherent in the generating function f. This general theory in conjuction with shape preserving aspects of the traditional splines provides algorithms for the construction of shape preserving fractal interpolation functions. Even though the results obtained in this article are generally enough, we wish to apply it on a specific rational cubic spline with two free shape parameters.