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 α-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 α-fractal function fα 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. © Springer India 2015.

Arya Kumar Bedabrata Chand

Department of Mathematics

Saurabh K. Katiyar

Constraint theory

interpolation

FRACTAL FUNCTIONS

FRACTAL INTERPOLATION

HERMITE

HERMITE INTERPOLANT

ITERATED FUNCTION SYSTEM

Positivity

SCALING FACTORS

SECOND DERIVATIVES

Fractals

IIT Madras is a public technical and research university located in Chennai, Tamil Nadu. Founded in 1959, it is recognised as an Institute of National Importance.

IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

It currently offers undergraduate, postgraduate and research degrees across 16 disciplines in Engineering, Sciences, Humanities and Management. About 596 faculty belonging to science and engineering departments and centres of the Institute are engaged in teaching, research and industrial consultancy.

IIT Madras

Quintic Hermite Fractal Interpolation in a Strip: Preserving Copositivity

Springer Proceedings in Mathematics and Statistics

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.

Calcolo

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

This paper is concerned with interpolation subject to a strip condition on the first order derivative using a class of rational cubic Fractal Interpolation Functions (FIFs). This facilitates the FIF to generate monotonic curves for a given set of monotonic data. The proposed monotonicity preserving rational FIF subsumes and supplements a classical monotonic rational cubic spline. In models leading to the monotonicity preserving interpolation problem wherein the first order derivative of the constructed interpolant is to be nondifferentiable in a finite or dense subset of the interpolation interval, the developed fractal scheme is well-suited in contrast to its classical nonrecursive counterpart. It is shown that the present fractal interpolation scheme has O(h4) accuracy, provided the original function belongs to C4(I) and the parameters involved in the FIF are appropriately chosen. © 2014 Elsevier Inc. All rights reserved.

Applied Mathematics and Computation

A fractal procedure for monotonicity preserving interpolation

Fractal interpolation function defined through suitable iterated function system provides a method to perturb a function f∈C(I) so as to yield a class of functions fα∈C(I), where α is a free parameter, called scale vector. For suitable values of scale vector α, the fractal functions fα simultaneously interpolate and approximate f. Further, the iterated function system can be selected suitably so that the corresponding fractal function fα shares the quality of smoothness or non-smoothness of f. The objective of the present paper is to choose elements of the iterated function system appropriately in order that fα preserves fundamental shape properties, namely positivity, monotonicity, and convexity in addition to the regularity of f in the given interval. In particular, the scale factors (elements of the scale vector) must be restricted to satisfy two inequalities that provide numerical lower and upper bounds for the multipliers. As a consequence of this process, fractal versions of some elementary theorems in shape preserving interpolation/approximation are obtained. For instance, positive approximation (that is to say, using a positive function) is extended to the fractal case if the factors verify certain inequalities. © 2014 Elsevier Inc.

Journal of Mathematical Analysis and Applications

Fractal perturbation preserving fundamental shapes: Bounds on the scale factors

The theory of splines is a well studied topic, but the kinship of splines with fractals is novel. We introduce a simple explicit construction for a L1-cubic Hermite Fractal Interpolation Function (FIF). Under some suitable hypotheses on the original function, we establish a priori estimates (with respect to the Lp-norm, 1≤p≤∞) for the interpolation error of the L1-cubic Hermite FIF and its first derivative. Treating the first derivatives at the knots as free parameters, we derive suitable values for these parameters so that the resulting cubic FIF enjoys L2 global smoothness. Consequently, our method offers an alternative to the standard moment construction of L2-cubic spline FIFs. Furthermore, we identify appropriate values for the scaling factors in each subinterval and the derivatives at the knots so that the graph of the resulting L2-cubic FIF lies within a prescribed rectangle. These parameters include, in particular, conditions for the positivity of the cubic FIF. Thus, in the current article, we initiate the study of the shape preserving aspects of fractal interpolation polynomials. We also provide numerical examples to corroborate our results. © 2013 Springer Science+Business Media Dordrecht.

BIT Numerical Mathematics

A constructive approach to cubic Hermite Fractal Interpolation Function and its constrained aspects

Quintic hermite fractal interpolation in a strip: Preserving copositivity

Journal	Data powered by TypesetSpringer Proceedings in Mathematics and Statistics
Publisher	Data powered by TypesetSpringer New York LLC
ISSN	21941009
Open Access	No