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. © Springer Nature Singapore Pte Ltd. 2017.

Arya Kumar Bedabrata Chand

Department of Mathematics

Saurabh K. Katiyar

K. M. Reddy

Blending

Data visualization

interpolation

Rational functions

CONSTRAINED INTERPOLATION

Convergence

FRACTAL INTERPOLATION FUNCTIONS

FRACTAL SURFACES

ITERATED FUNCTION SYSTEM

Positivity

Fractals

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IIT Madras

Constrained Data Visualization Using Rational Bi-cubic Fractal Functions

Communications in Computer and Information Science

One-sided approximation tackles the problem of approximation of a prescribed function by simple traditional functions such as polynomials or trigonometric functions that lie completely above or below it. In this paper, we use the concept of fractal interpolation function (FIF), precisely of fractal trigonometric polynomials, to construct one-sided uniform approximants for some classes of continuous functions. © 2016 World Scientific Publishing Company.

FRACTAL TRIGONOMETRIC POLYNOMIALS for RESTRICTED RANGE APPROXIMATION

Fractal interpolation provides an efficient way to describe data that have smooth and non-smooth structures. Based on the theory of fractal interpolation functions (FIFs), the Hermite rational cubic spline FIFs (fractal boundary curves) are constructed to approximate an original function along the grid lines of interpolation domain. Then the blending Hermite rational cubic spline fractal interpolation surface (FIS) is generated by using the blending functions with these fractal boundary curves. The convergence of the Hermite rational cubic spline FIS towards an original function is studied. The scaling factors and shape parameters involved in fractal boundary curves are constrained suitably such that these fractal boundary curves are positive whenever the given interpolation data along the grid lines are positive. Our Hermite blending rational cubic spline FIS is positive whenever the corresponding fractal boundary curves are positive. Various collections of fractal boundary curves can be adapted with suitable modifications in the associated scaling parameters or/and shape parameters, and consequently our construction allows interactive alteration in the shape of rational FIS. © 2014, Springer-Verlag Italia.

Calcolo

Positive blending Hermite rational cubic spline fractal interpolation surfaces

This article explores the properties of fractal interpolation functions with variable scaling parameters, in the context of smooth fractal functions. The first part extends the Barnsley-Harrington theorem for differentiability of fractal functions and the fractal analogue of Hermite interpolation to the present setting. The general result is applied on a special class of iterated function systems in order to develop differentiability of the so-called -fractal functions. This leads to a bounded linear map on the space which is exploited to prove the existence of a Schauder basis for consisting of smooth fractal functions. © 2015 Australian Mathematical Publishing Association Inc.

Bulletin of the Australian Mathematical Society

Fractal bases for Banach spaces of smooth functions

Fractal interpolation is a modern technique in approximation theory to fit and analyze scientific data. We develop a new class of C1- rational cubic fractal interpolation functions, where the associated iterated function system uses rational functions of the form (Formula presented.) where pi(x) and qi(x) are cubic polynomials involving two shape parameters. The rational cubic iterated function system scheme provides an additional freedom over the classical rational cubic interpolants due to the presence of the scaling factors and shape parameters. The classical rational cubic functions are obtained as a special case of the developed fractal interpolants. An upper bound of the uniform error of the rational cubic fractal interpolation function with an original function in C2 is deduced for the convergence results. The rational fractal scheme is computationally economical, very much local, moderately local or global depending on the scaling factors and shape parameters. Appropriate restrictions on the scaling factors and shape parameters give sufficient conditions for a shape preserving rational cubic fractal interpolation function so that it is monotonic, positive, and convex if the data set is monotonic, positive, and convex, respectively. A visual illustration of the shape preserving fractal curves is provided to support our theoretical results. © 2013 Springer-Verlag Italia.

Shape preservation of scientific data through rational fractal splines

This article introduces fractal perturbation of rational functions via α-fractal operator and investigates some approximation theoretic aspects of this new function class, namely, the class of fractal rational functions. Its specific aims are: (i) to define fractal rational functions (ii) to investigate the optimal perturbation to a traditional rational approximant corresponding to a continuous function (iii) to establish the fractal rational function analogues of the celebrated Weierstrass theorem and its generalization, namely, the Müntz theorem (iv) to prove the existence of a best fractal rational approximant to a continuous function defined on a real compact interval, and to study certain properties of the corresponding best approximation operator. By establishing the existence of fractal rational functions that are copositive with a prescribed continuous function, the current article also attempts to invoke fractal functions to the field of shape preserving approximation. © 2014 Elsevier Inc.

Fulltext

Journal of Approximation Theory

Fractal rational functions and their approximation properties

Constrained data visualization using rational bi-cubic fractal functions

Journal	Data powered by TypesetCommunications in Computer and Information Science
Publisher	Data powered by TypesetSpringer Verlag
ISSN	18650929
Open Access	No