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.

Arya Kumar Bedabrata Chand

Department of Mathematics

Saurabh K. Katiyar

Computational mechanics

functions

Identification (control systems)

interpolation

Polynomial approximation

APPROXIMANTS

Continuous functions

FRACTAL INTERPOLATION FUNCTIONS

ONE-SIDED APPROXIMATION

Trigonometric functions

TRIGONOMETRIC POLYNOMIAL

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

Fractal trigonometric polynomials for restricted range approximation

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.

Communications in Computer and Information Science

Constrained data visualization using rational bi-cubic fractal functions

Acta Applicandae Mathematicae

Bernstein Fractal Trigonometric Approximation

In this paper, we explore the Riemann–Liouvllie fractional calculus of quadratic fractal interpolation function (QFIF) with variable scaling factors. Fractional calculus of QFIF with predefined initial condition is investigated in an arbitrary closed interval of R. Further, the relation between the order of fractional integral (derivative) and the box dimension of QFIF is established. © 2018, Forum D'Analystes, Chennai.

The Journal of Analysis

Riemann–Liouville calculus on quadratic fractal interpolation function with variable scaling factors

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

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

Fractal interpolation functions provide a new light to the natural deterministic approximation and modeling of complex phenomena. The present paper proposes construction of natural cubic spline coalescence hidden variable fractal interpolation surfaces (CHFISs) over a rectangular grid $\mathscr{R}$ through the tensor product of univariate bases of cardinal natural cubic spline coalescence hidden variable fractal interpolation functions (CHFIFs). Natural cubic CHFISs are self-affine or non-self-affine in nature depending on the IFS parameters of univariate natural cubic spline CHFIFs. An upper bound of the error between the natural cubic spline blended coalescence fractal interpolant and the original function is deduced. Convergence of the natural cubic CHFIS to the original function $f \in C4[\mathscr{R}]$, and their derivatives are deduced. The effects free variables, constrained free variables and hidden variables are discussed on the natural cubic spline CHFIS with suitably chosen examples. © 2012 World Scientific Publishing Company.

Natural cubic spline coalescence hidden variable fractal interpolation surfaces

Fractal interpolation functions with variable parameters and their analytical properties

FRACTAL TRIGONOMETRIC POLYNOMIALS for RESTRICTED RANGE APPROXIMATION

Journal	Fractals
Publisher	World Scientific Publishing Co. Pte Ltd
ISSN	0218348X
Open Access	No